Gettng around Edinborough

There are no flights between Newcastle and Edinburgh, but there are frequent trains. There are many railway stations in and around Edinburgh, but the two big ones in the centre are Waverley and Haymarket. Waverley being the larger of the two and probably considered the main station in Edinburgh. It's also the closest to most attractions.

Which station to get off depends on where you are staying. As Badger stated, Waverly is the "main" station but Haymarket can be easier to manage if staying near it. For most, Waverly probably makes the most sense.

You want to take the train from Newcastle to Edinburgh because much of that trip is along the coast and very pretty.

For your tours, take a look at Rabbies

Edinburgh Park and Edinburgh Gateway would be a change to a frequent local train at Waverley, as would Haymarket on the majority of trains from Newcastle (a few do go through to Haymarket).

Both Edinburgh Park and Gateway have very good value budget hotels close by, and Gateway has the trams to the airport and into the city centre. Both are very convenient for the City Centre. Edinburgh Park also has buses to the Airport.

Our resident transportation gurus have given you their expert advice.

To address your other questions..,
For your time in Edinburgh, you say you “wish to visit the museum, the Kelpies, and take a 3-day bus tour including Isle of Skye.”

Which museum? I’m guessing you mean the National Museum of Scotland, but there are many other museums of note in Edinburgh. All are easily accessible on foot or by bus.

I assume you know that the Kelpies are not in Edinburgh but are a stop on various day trips you can take from the city, such as Rabbie’s Loch Lomond National Park, Kelpies & Stirling Castle 1-day tour.

Rabbie’s also has multi-day tours that include Skye, as do other tour operators.

How many days have you planned to spend in Scotland after visiting your friend in Newcastle?

Looks like a train from Newcastle to Edinborug Waverley. Is it feasible for me to walk to a hotel in that area that is safe? then to the pick up for a tour bus. Looking at Highland Explorer Tours to Isle of Skye, 3 day. Then back to a hotel and a day tour to the Kelpies before I train back to Newcastle to fly home. Is the area around Waverley safe? No homeless encampments, etc....? Total time in Scotland 7 days max.

Edinburgh Waverley is right in the centre of the city and is perfectly safe. You may well see homeless people - there are a lot of them around in the UK - but I wouldn’t categorise them as a threat.

Plenty of hotels in the area. I’m not sure what your budget is, but there’s something to suit most people.

Have you already booked your flights? If not, I would suggest doing a multi-city booking, flying into Newcastle and out of Edinburgh. That would save you the return train journey to Newcastle at the end of your trip.

Why feed the troll? The op is clearly having a laugh.
Just in case she isn't, Edinburgh city centre is safe enough. There are one or two suburbs she probably wouldn't like but the average tourist is highly unlikely to wander into them.
Of course you can't fly between Newcastle and Edinburgh. You have the option of a train, bus or coach. Trains are expensive and prohibitively so on the day but it's a nice route and only takes about an hour and a half. Coaches are about 2 and half hours. Try National Express, Flixbus or Megabus.
It used to be possible to get a local bus via the borders once a day from Newcastle to Edinburgh up through Northumberland. I think you still can but you'd probably have to change at Jedburgh.
Given that the op appears to think you can fly between Newcastle and Edinburgh, I'm a bit concerned she's under the impression she'll be doing day trips to Skye from Edinburgh....

Take the train from Newcastle to Edinburgh Waverley, which is right in the centre.
Stay in a hotel within walking distance of that station....look on Google maps for where those are.
Edinburgh is more than safe for visitors.

Book your tour to Skye well in advance, including accommodation, which is probably starting to get scarce even now so early in the year.

Also book your tour to see The Kelpies.
No car....easier to take a tour to there.

Fly home from Edinburgh...easy to get to the airport by bus or tram from near or at Waverley Station.

Don't go to Edinburgh in August...it's super busy during the Festival...unless you get booking everything there very, very soon.

(Also, the correct spelling is "Edinburgh".)

Why feed the troll? The op is clearly having a laugh

There's no stupid questions here. Many have no experience of European travel at all.

Welcome to the forum.

Looks like I’ll be flying into Newcastle. Then after my visit, taking a train or bus to Edinburgh. Stay in a hotel in the area. Visiting the Natural Museum and the city. Then taking the 4 day tour available to the Kelpies, Jacobite Steam train ride and Isle of Skye. At the end flying from Edinburg back home.I see one of the tour companies include the Kelpies on a 4 day.

Some thoughts: Newcastle is the next big city from Edinburgh on the main railway line to London. There are fast trains at least every hour and train is the only sensible option for travel between the two places for someone with no car. Those trains terminate at Waverley, except the ones that continue to Aberdeen which will also stop at Haymarket. But often the train waits for a while at Waverley even if continuing north. So unless you have a well thought out plan to the contrary, you're going to get off at Waverley. You're now in the center of Edinburgh. There are many hotels around of course, but typically quite full/high priced. So accommodation further out of town might be a better option depending on your budget/personal preferences. From Waverley to your accommodation I'd recommend take a taxi because navigating Edinburgh busses/trams with luggage is not always pleasant. After you get installed at your accommodation you can walk/use busses to get around. Nowhere a tourist is going to go in Edinburgh these days is unsafe but use the same street awareness you'd have in any large city, especially at night. One thing to note regarding your plans is that the Kelpies are not far from Edinburgh so you could decouple that stop from your Highland bus tour and perhaps simplify things -- you can run out one evening to the Kelpies from Edinburgh. Remember that in summer when most tourists go it is daylight until 9pm or later, so you can fit in outdoors things into a long day. Plus the light for photography is better late in the day. For example the Kelpies car park closes at 10pm in the summer. Take the train to Falkirk then use Mr Uber or taxi, or perhaps persuade an Uber guy to drive you all the way for a reasonable fare. It's a 45 min drive. Blackness castle is also interesting out the same way if you're into Outlander.