If you are interested in seeing some contemporary art at the high end of the market, be sure to visit the galleries in Cork street and the parallel streets (through Burlington Arcade, which is next to the Royal Academy)
There is usually some really good work on show (for sale there of course as these are the commercial galleries - quite handy if you have the odd £28,000 or so to spare :wink: ), Whenever I go down for a big show at one of the public galleries I usually try to get to the Cork street area as well.
If you google Cork Street and galleries you'd be able to find out what's showing. The Messum gallery is one of my favourites.
Back to the Tate - I loved the Rothko's but was a little disappointed in a lot of the work as a lot is from an era of British art that I don't much relate to. Special exhibitions (like the Kahlo) can be great but I haven't been to see that so can't comment on it.
The private galleries have a booklet that you can pick up with maps and details of work showing at hundreds of galleries in London - really useful and free.
when I went to the Tate last the nice new bridge was in use (it hadn't been for a while because it swayed if too many people were on it at once if you remember! ) but they hadn't put lights on it for shipping and so because of an ancient bylaw with its height there were bundles of hay suspended at intervals, hanging down over the river to warn them! very high tech
have fun