I just returned from a 2.5-week journey to the UK (and 4 days in Paris), with Edinburgh as one leg. I loved Scotland! I would have loved to explore it even more; we basically had 4 nights, and spent 3 of the days exploring Edinburgh, only to find we'd wanted more time; we had an additional day on a Rabbie's tour (lochs, castles & highlands-- beautiful country, so yes, take your time exploring the rest of Scotland!).
You mention that "there is so much to do in so many places, I'm overwhelmed trying to pick!" This is the story of my travels, and most other people's, and in order to mitigate any regrets, I've tried two things...
- Choose 1-2 things I really, really want to see or do (perhaps a 3rd as a substitute in case weather makes one of them iffy)
- Allow time for exploration of 2-3 other things that are close together, with an eye towards exploring that area/region in the process to see what you stumble on
- Allow a chunk of a day just to wander aimlessly, allowing for serendipitous discoveries (possibly my favorite memories off the bat); even if nothing big happens, that amble can allow you just to soak in what's around
My list of things I'd wanted to do that I didn't get to this round is longer than the list of things I did, and that's just always going to be the case! But on our ambling day, after a beautiful stroll through the Royal botanic Gardens, we thought we'd wander into the National Museum of Scotland for an hour (so I could see my adored Lewis Chessmen), and it turned out to be a wonderful place with a lot going on (especially fun for our teenage daughters). We were there 3 hours and still didn't get to everything.
That said, 3 days in Edinburgh will give you plenty of time to enjoy it. It's a charming city, and while London has been my first love, Edinburgh now rivals it, and in comparison is more navigable and a bit easier to get your mind around! Enjoy!