Tate Britain and Modern Museums Vs. the National and Portrait Gallleries

All four are world class museums and all worthy of a visit. If you need to prioritize research the temporary exhibits running during your stay, which ever ones peak your interest the most let that be the deciding factor.

Or base it on your interests in art. I wouldn't miss the National Gallery. The Portrait Gallery is really about history, all the people whose faces you see. There are terminals where you can look them up to put them in context. Tate Britain is good, much smaller than National Gallery and farther from other sights. Tate Modern is a fascinating building, not my cup of tea re the art but could be yours!

Unless your interests center on modern art, I'd say the National Gallery should be on your list.

You could go to the website of each museum and get more information. And then decide based on your interests and preferences.
A couple of weeks ago, I did exactly what you're thinking: went to two of them: the National Gallery and Tate Britain.
On prior trips I've been to the Tate Modern and Portrait Gallery.

Tate Modern is modern art.
Portrait Gallery is portraits of people in UK history.

It has been said that Tate Britain has the best collection of Turner's in the world, also many by Constable; it focuses on the best known British artists.

The National Gallery covers all periods of art, they say it's one of the top comprehensive art museums in the world. It's like the British Museum in that you cannot "do" it in one visit. One strategy is to pick your favorite period or two of art and focus on those.

I hope the above helps.

IMO two of the four would be good, just as you're thinking. Which two depends on your interests and preferences.
You could read Rick's descriptions in his book London 2014 and/or listen to his guided tours of these museums.

Maybe get a taste of both ends of the art world by going to the Tate Modern and the National Gallery? Both can be seen in 1/2 a day if you don't linger too long in any one area. I could spend days in each but you can get a taste if you just pick a section or two in each and go for several hours. The Tate Modern has numerous galleries, some with permanent exhibitions and others that are rotating. Some of the temporary exhibits require a fee but most are free. Exhibits are edgy, extremely modern and experimental, and much of the work is displayed in very interesting ways. Go with an open mind! Their gift shop is worth a visit - it's extensive and interesting. The self-service cafe has a spectacular view of the Thames and St. Paul's Cathedral.
The National Gallery is also easy to digest if you choose the parts you are most interested in and spend a morning or afternoon. It is divided up by styles and eras so pick what you like and have at it! There are very well-known works that you will probably recognize and quite a few others that are lesser known but equally wonderful. The Execution of Lady Jane Grey almost brings me to tears every time I see it...

If you have any interest in Impressionist art, the National Museum has a wonderful collection. Even in only an hour or so you can see some of the best of the best in that gallery. It's a free museum (still, I think) and in an area where you are likely to find yourself. The Portrait Gallery is just around the corner. There is always a special exhibit going there on the ground floor, and it's usually exceptional. Special exhibits require a paid ticket, but the regular collection is free. Take the elevator up to the top, start walking, and stop at what interests you. Lots of variety here, from old Tudor portraits to a wonderful big splashy painting of Paul McCartney, and of course, portraits of the current royals. Bet you'll find something you like!

"It's a free museum (still, I think) ."
Yes, it is still free. They ask for a small donation, which is optional.

"...Portrait Gallery is portraits of people in UK history..."

The museums has gotten a bit more diverse over the years. They now feature photography of notable celebrities like super models, musician, soccer players etc.

Tate Britain is specifically for British art, both historic and modern. The only issue I'd point out is that Tate Britain is not particularly convenient to get to - the only really convenient route is by bus route 87 from Trafalgar Square or Westminster.

Thank you all so much for the great advice! I think that I will choose the National Gallery and the Tate Modern and save the Tate Britain and Portrait Gallery for another trip. Thanks again for helping me sort this out!

I went to both Tates and the Portrait Gallery and I must say I loved the Portrait Gallery the most, arranged historically and full of so many fascinating people. Next favourite would be the Tate Britain, though I do like modern art some of it is just odd. Most interesting installation was Ai Weiwei's Sunflower Seeds.

If you are a fan of Pre-Raphaelite art, Tate Britain has an excellent collection of some of the most famous paintings in that genre.

Also, you can take the Victoria line tube to Pimlico and follow the signs to Tate Britain.

Any two of those museums will give you a memorable experience without overload. It's really all about your personal interests.

For me, the Tate Modern stood out as my top site in all of London. Its architecture is interesting and a perfect match for the art it presents. I do like modern art, not all of it, some leave me cold, some spark controversy but some pieces touch me on a very personal level, more than any other period in art history. I'm also a fan of Joseph Beuys and the Tate Modern has a number of good installations with very good commentary.