In terms of travel to and from, we live in Boise, ID, so we always need to reposition. Our most frequent routes are a seasonal route on United from BOI thru ORD to EDI and BA through LHR to EDI. The United route seems to be a summer seasonal flight. Lately, we've been taking BA almost exclusively either from DEN or DFW through LHR to EDI using BA Avios or Qantas points to book BA flights. We have done the train a number of times in both directions (Edinburgh to London and vice versa) if we were only able to find biz class seats to and from LHR or a couple of times when our EDI to LHR (or vice versa) leg was either cancelled or changed to a time that wouldn't work with the transatlantic leg. The train ride does add anywhere from 4.5 to 6 hours to the journey depending upon whether you get an express train (and the logistics of getting to/from the train stations), but the train ride itself is very pleasant and quite scenic once you get into Northern England and Scotland. On my last trip to Edinburgh in August/September, I deliberately scheduled my return from LHR so I could spend a couple of nights in London on my way home and see Six. (I also happened to be in Scotland when the Queen died, and that was such a fascinating piece of history to be a part of. I was in Edinburgh when they moved her body to Holyrood and St. Giles, and I was able to watch the procession of her hearse down the Royal Mile to Holyrood and also the live proclamation of the Queen's death and Charles' ascension to the throne. Then I was in London just ahead of her body arriving there.)
In terms of times of year to go, I have been to Edinburgh in every month except for March, July, October and November, and each season has something to offer. In April, you get the cherry blossom which is gorgeous. I was there for snow in Feb. which was spectacularly beautiful. I might advise that a first-time visitor to Edinburgh not go in August which is festival season. There are fun things to do then (like the Royal Edinburgh Military Tattoo which I highly highly recommend, and the Fringe and International festivals), but the city is insanely crowded at that time. I was there this past August visiting my daughter, and we did the Tattoo (for the third time, it's that good and changes every year) and a number of the festival shows. We really enjoyed it, but we had also been there twice previously in August when we were more focused on seeing sites, the university for my daughter, etc. and found ourselves more annoyed than anything at the crowds and how taken over the city was by festival venues. But a teen might really enjoy the festival vibe and the Tattoo, so you'll have to be the judge. In September, you will get heather, which is also beautiful, but beware that it comes with midges in the Highlands.
I haven't stayed in a hotel in Edinburgh for a couple of years since we now own a flat there, but we've stayed at both the Radisson Blu Edinburgh and the Hilton Carlton Edinburgh (both on the Mile), and I would recommend both. We also did an AirBnB in the Bruntsfield neighborhood where we ended up buying our flat, and that worked out really well too.
In Edinburgh, I'd recommend touring Edinburgh Castle and Holyrood Palace, hiking up Arthur's seat, strolling the Royal Mile and taking time to poke around some of the closes, St. Giles Cathedral, Dean Village and the Stockbridge neighborhood, Circus Lane, the National Museum of Scotland, the Writer's Museum, Princes Street Gardens, Gladstone's Land, Calton Hill, the Scott Monument, and I'm sure I'm forgetting a ton. I've hiked the West Highland Way on a group tour (just this past May with
@CaliforniaGirl09), and have also visited the Highlands on an
Adventures by Disney trip, but haven't yet done it on our own, so have less advice there. You could certainly take the train to Fort William or Inverness and use that as a home base for exploring the Highlands. And, as I mentioned in my prior post, Rabbie's does a day trip to the Highlands from Edinburgh. Other Rabbie's day tours that we have done and enjoyed are: the Outlander castles, St. Andrews and Fife, Roslyn Chapel and the Borders (that included Melrose Abbey), and another Borders tour that included Jeburgh Abbey and Hadrian's Wall. On my to do list are the Jacobite train up in the Highlands, and a three-isles tour from Oban that includes puffin viewing from late spring through the summer. We've also been to the Orkneys twice, and I would recommend, but that might be harder to fit in to an overall first-time Scotland trip. The Shetlands are also on my future must-do list.
Hope that's helpful. I'm definitely happy to answer specific questions.