Airfare

This doesn't require a travel agent, but it does require some research and patience. Go to rome2rio.com and check out the various options traveling between the three locations you list. In some cases buying a cheap internal flight is more efficient, in others train may be. Most on this site will tell you to fly "open jaw" - that is in to one city and home from another - it avoids back tracking and saves you time. It might not be cheaper though, but you'll have to evaluate that in the overall cost of the trip. Use www.google.com/flights and choose "multi city" from the first drop down box. Plug in PHX as your start and ending locations - e.g., PHX-Prague, then Budapest - PHX. Play around until you find options that suit your needs. If you aren't set on dates, adjust those to see if prices work better for you.

What do you consider decent airfare?
Sometime it will be a lot cheaper to fly roundtrip out of the same two cities versus into one city and out of another
For example search Phoenix to Prague roundtrip or Phoenix to Prague Budpest to Phoenix,

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Define decent? Without some dollar amount it is hard to know what you are looking at. I think anything under 1200, maybe 1500 is decent. Others expect sub $800. Second, I have never found that an open jaw ticket (in one airport, out another or multi-city) is more expensive -- meaning $100 + or -. Most of our trips are open jaw as we find that more efficient. AND -- most importantly - WHEN??? Makes a lot of difference. May is a long month. The end is near the beginning of the summer.

Condor Airlines is showing a direct flight to Frankfurt from Phoenix. Perhaps fly round trip to Frankfurt on Condor and than use low cost carriers or trains for the remainder?

I took a preliminary look on Google flights, and for a 2-week trip during May (no specific dates), I found very little difference between round-trip costs to Prague and to Budapest. That doesn't necessarily mean that a multi-city flight into one of those cities and out of the other would be about the same cost; you have to try it out for your specific dates to see. And you need to consider which flight routings are acceptable to you. It doesn't matter if you can find a fabulous deal if you'll be flying for 24 hours and you're not willing to do that.

In my analysis Krakow round-trips were several hundred dollars more expensive than those to Budapest and Prague, but you'll need to fly in and out of Krakow in any case. You may save money by putting Krakow in the middle of your itinerary, but it will be at the cost of having two mid-trip days when you have to deal with airport hassles going to and from Krakow. If you start or end in Krakow, you can travel by train between Prague and Budapest (6-1/2 hours) if you want to and have only one flight day in the middle of your trip.

To me, it would be worth something extra to start or end the trip in Krakow, but not if flying to Krakow was going to require an extra stop and many more hours compared to Budapest and Prague.

Those are all great cities, but your trip would be easier to arrange if you postponed Krakow until your next trip to Europe. It is far from the only interesting destination in Poland.

We always use Google Flights and Kayak to see what flights are available. You want "multi city" option so you don't have to return to first stop in your itinerary, saving time and money.
From PHX, fly to LAX, DEN, ORD or DFW to connect.

How much time do you have for this trip? Enough to really do all three?

With the triangle of cities you have chosen, I would be looking at 'open-jaw' ( aka multi city) of into one out of another and put the 3rd location in between.

Its just going to be a matter of 'trial and error' until you find what works for your dates and your budget

We did a similar trip to this in 2017 although we flew into Warsaw for three days before heading to Krakow. We took an afternoon train between Warsaw and Krakow which gave us a peek at what the Polish countryside looks like. Seeing that gave us a good idea of why our ancestors settled here in the midwest. Since it looked so much like the midwest, they probably felt at home here. We did Warsaw on our own and hired a private driver in Krakow who gave us a tour of the city and the next day took us to tour the salt mines and on to Auschwitz/Birkenau. Our driver did offer transportation to Prague, but we chose to take the overnight train to Prague feeling that we would be saving time and reenergizing by sleeping in a compartment on the train. I had asked for advice on this forum about the safety and comfort of making the trip by train and most advised against it. But we had previously taken an overnight Amtrak trip here in the US and enjoyed it. So we decided to go for it hoping to save time while being able to rest on the way. While I wouldn't recommend sitting up all night, the compartment offered a private place to sleep. We arrived in Prague at 6:30am rested and ready to start the Best of Prague and Budapest tour. We made our way to the hotel where we were staying at the first of the tour and they were happy to hold our luggage until check in time. We set off on foot, found a lovely place for breakfast and had the day to familiarize ourselves with the city before meeting our RS tour mates. We did fly home open jaw at the end of the tour as it made more sense than backtracking. Hope you have a great time! Central Europe is a great place to visit!

Can you make a list of all the possible flight plans available that might possibly be of interest to you? As you get a sense of what flight plans are too complex, or irrational because they were made by a computer, make another list with the flight plans that seem most appealing. Then you will see which flight plan has the best combination of least time and lowest price. Then perhaps ask us which of those few flight plans we would pick. It does look an “open jaw” trip would require two major ground trips, versus 3 major ground trips flying in and out of the same city. I have only taken two major trips so far that involved seeing more than one city; in Italy I flew to and from Rome; in Greece I flew to and from Athens.

As a footnote, I flew round trip to a major European airport and then took other flights as such was far cheaper than an open jaw ticket and the alternate itinerary worked well. If the price spread is large, I can be bought. The http://matrix.itasoftware.com/ airfare website calendar function works well with various results for consideration. Happy research.

Separate intra-European hops (by air or by train) to get from the cheap landing point to the first planned destination or from the final planned destination to a cheap departure airport can save money. But the practicality of that technique depends partly on how long the trip is. Those extra hops can cost the better part of a day, because you can't schedule yourself very tightly. That lost time is borderline-unbearable on a short trip. We don't know how much time Meredith has for her trip to these three lovely cities, each of which is worth a minimum of four days (5 nights).

Condor Airlines is showing a direct flight to Frankfurt from Phoenix. Perhaps fly round trip to Frankfurt on Condor and than use low cost carriers or trains for the remainder?

I second this with great emphasis. This is without doubt I think the best option overall. The trains will be fast and comfy and returning to Frankfurt is an easy thing to do from the cities you listed...best of all on the return you land at home.

Time is money, and flying into a city you don't want to visit will cost you vacation time commuting to where you actually want to arrive. The suggestion of matrix.itasoftware.com is good advice. Although owned by Google, it does not sell tickets. But it also, in my tests, offers more choices than Google's Flights.

Always use a multi-city search function to shop for this sort of itinerary. It will cost you less time to go to an agent, if that agency is large enough to have ties to specific hotels. But agencies get no flight bargains, not even the big on-line agencies. Use their facilities for data and then go to the airlines' own sites.
The only time I book through Expedia or CheapoAir or so many others is for a complicated itinerary where the airlines don't sell each other's tickets.

Re: Flying to a major European city, then taking short flights to your destination.

Another factor to consider is whether you would like to visit the arrival city. For example, we went to Sicily a couple of years ago, and instead of flying there on one ticket, we stopped over in Rome before and after. Since Rome is one of our favorite cities, the stopover was fun.

IMHO a travel agent is never a good idea. It's not that I hate travel agents, I just think they don't know your preferences as well as you do. Is it worth $200 extra to save two hours? For some people the answer is yes, for others it's no. A travel agent will guess because they don't know.

From Phoenix, I'd find any good airfare/route to any major European city (there used to be a direct flight to London). After that, get a cheap hop on a European carrier to where you want to go. I'd give myself plenty of time (for me at least 24 hours) at your first stop to give you some room for delays and connections. Scheduling a short turnaround is rarely a good idea, but really not a good idea when it's on different tickets - not showing up for your flight means finding and paying for a new flight is on you.

When I buy tickets I start with a search. I look at alternate airports on both ends as well as adjust dates to see if I can save money there. I save a number of searches and watch the price for awhile. Ultimately, I want to know all my options and prices for those options. From looking at different options over a month or so, I then have an idea what a decent/good price is for my flight. From there I keep watching flights until one of two things happens, I see a great price (maybe a fifth or more lower than a decent price) or the seats for good routing at the decent price are nearly sold out. Then I book.

I'll pay extra for a eight hour duration to Europe over an eleven hour duration with a connection. How much extra? It's hard to say, I have to see the options and weigh it out for myself. I rule out all the super cheap options where you fly to Moscow or Istanbul as the first leg, wait ten hours, then backtrack to your destination. To me those aren't good deals - but they might be seen as a great idea by someone else.