Zurich to Zermatt to Interlaken

1) There are reasonably priced hotels near the airport (not at the airport).
Ibis, Dormero, amongst others. Can take the bus or tram to get to them.

2) I am not sure there is any budget option apart from a hostel in Zermatt.
This is overall, IMO, the most expensive ski resort in all of Europe (well,
maybe St Moritz).

I assume you are aware that you've picked a non-prime part of the year
to ski. While some runs may be open, getting to them may involve a lot
of uploading and also downloading at the end of the day, and conditions
could be quite variable.

3) The lift infrastructure in Zermatt is actually not all that great and it will take
time to get to the higher elevations where there is still snow. I would not
expect you to be able to get more than a few runs from when the lifts
open to when you have to come back down, change, pack, and leave before
you get charged for another day of lodging (since you're on a budget). The train
schedule is not the issue, so if you can get a late checkout time (or be OK
with leaving your luggage at the front desk and changing in the hotel bathroom)
this can probably work.

What is a budget? I would start with booking.com to get an idea, both at Zurich airport and Zermatt.
But getting to Zermatt the day before is a good idea so you will have equipment and clothing to get an early start. Like I said, probably by 1 pm or so, it will be time to head down, and head to Interlaken. If you can depart Zermatt by 4 pm, you can be in Interlaken in 2 1/2 hours, or 3 1/2 if you want to take a scenic route.

With only 3 days, 3 teens and jetlag you need to pick 1 spot - Zermatt OR the Jungfrau area. Trying to do all of this is going to be nuts. I personally wouldn't be messing with skiing at that time of year, but my guess is that's a non negotiable for your group, so just go to Zermatt. If you try to pack in 3 different hotels plus the transportation while you're on a budget, you're going to be in for a shock + you're going to spend half of your short amount of time dealing with logistics instead of actually enjoying the mountains. Since skiing sounds the highlight for you all, just go to Zermatt the whole time and then head to Germany. You're going to be surprised to find how few places even remotely resemble "on a budget", especially since you're so close to the time. Anything that maybe was in budget was long ago rented.

1 - Zurich Airport (you can eat dinner in the mall at the airport and then just go to bed to have any chance of getting up and going at a reasonable time the next morning with jetlagged teens)
2 - Zurich > Zermatt - explore the village
3 - ski
4 - Gornergrat/hiking
5 - Zermatt > Frieberg

Thank you wanderweg for the consideration of our ideas.  We may be opt to skip the summer skiing and just go to Interlaken instead... or ONE other area that has the most WOW for our first trip.  Is that a better plan? I was to get a room at Interlaken youth hostel for our dates so that is fantastic!  

How does this look:
1) Zurich
2)Interlaken and day trips
3)Interlaken and day trips
4)Interlaken and day trips or head out to Germany

Or we could choose another area to stay in if there is a better idea?!? We like a lot of activity and excitement and can't wait for this adventure!!!!

Is the Zermatt plan better for teens?

shoeflyer thank you for the info I didn't think about some of the things you mentioned such as mountain conditions and travel to the top. I will check out the hotels you sent thank you!

We may skip skiing but need an alternate plan!!! AH! The time is ticking

Since you are going in June, book your hotels ASAP. June is no longer low season.

Thank you barkinpark I'm currently looking for hotels. It is taking a lot of time and I'm only on the first night out of a 14 night trip! It looks like I had better keep the reservation for Interlaken that we already have. Travel with 5 people isn't so easy but we will do what we have to do to make it work! :-) I might be up all night searching.

Any tips on web searching would be great right now. I'm using google

Wish me luck and thank you for the reply!

Use Google Maps for the Zurich Airport area, then key in hotels and the page will allow
you to add in your dates and # of people (obviously, you may need more than 1 room).

You will then get rates for all of the hotels in the page, and you can scroll in/out to
narrow or widen the area you see.

This works great in cities, but is less effective in mountain regions where elevation
and topology are not as easily seen and you might book a place farther away then you
want to be from whatever is important to you. But it still can give you an idea of what
you're up against.

I used booking.com and I saw aa apartment for 5 in Kloten for ~360 CHF. That was the best deal going.