Lots to do in Edinburgh - its pretty and very tourist friendly. As the capital it is where you can find the national museums and art galleries, and if you are interested the parliament is at the bottom of the Royal Mile (the high street which runs from the castle through the old town). There is lots to do but you can easily just wander around, visit little cafes, walk across the meadows/up Arthur seat and just take a relaxed pace of life. You can go to Fife and St Andrew's pretty easily from Edinburgh too.
Glasgow likes to think itself as a friendly place (though this could be disputed!), but is bigger and less pretty. There are a few museums and things but its a bit more spread out. Glasgow is the louder, pushy brother!
Stirling is a historic place - a good castle and the near Bannockburn. Linlithgow is a nice little place to visit in between Glasgow and Edinburgh.
Inverness is a long journey from the central belt - a nice journey but the city itself isn't particularly worth a visit, the mountains and outdoor sports with the scenery is the main attraction of the Highlands. There is also a lot of culture in terms of local art and music.
Weather varies but it rains every day in Glasgow and in general I would expect it to be cooler than in Leeds.