Welcome aboard, adblue!
GURPS is an enormously flexible and versatile RPG. Even without supplements it can probably handle all the content of the genre and style of adventure that you indicated — a bit of Indiana Jones and a bit of Doc Savage and and bit of Captain Nemo and a bit of the Phantom and a bit of Tarzan and a bit of Mandrake the Magician. It is almost certainly a good system for the setting.
There remains, however, the question of whether
GURPS is a good match for your style of play.
- GURPS tends to be a bit detail-oriented, which is fine if you like that sort of thing (we wouldn't still be here unless we did) but not for everyone. One of the most important things you learn to do when setting up a campaign in GURPS is to decide which areas to treat with the detail of optional rules and which areas to gloss over a bit with the reduced detail of the basic rules and detail-reducing options. Don't try to focus on everything at once! If you want to play the sort of game in which a character has "a revolver" don't weigh your game down with the rules that distinguish between a .455 Webley and a Smith & Wesson .44-40. It's all there. Don't use the bits that won't help your game.
- Some games are designed to determine which player should get to choose what happens next. Some games are designed to make the course of an adventure turn out as though it were written by a TV-script hack, or by the Coen Brothers, or by H.P. Lovecraft. Some games are designed so as to produce results that are dramatic. Some are designed to present the players with an interesting tactical challenge. GURPS is designed to simulate alternative realities. If you ask GURPS "what would happen if a man of Bronze twice as strong as an Olympic sprinter swung a demon's double-sized sword at a centaur armoured with NIJ Level IIIa barding?" it will answer that question, not give you the most dramatic result, the result that R.E. Howard would have written, or a chance for a player to narrate how his character achieved a decisive victory.
I would suggest that the
GURPS core —
Characters and
Campaigns — probably contains everything you
need for the setting you described. In fact, it contains a lot more than you need, because it covers many other settings as well. There are a lot of supplements that might be a lot of fun to add in —
GURPS Martial Arts if you want to go into detail with hand-to-hand fighting (and not just the well-known Asian martial arts we habitually think of),
GURPS Gun-fu or
GURPS Tactical Shooting,
GURPS Horror,
GURPS Action, even the 3rd-ed product
GURPS Cliffhangers, which is still very useful. But you don't have to jump in with both feet to get started. I would suggest that you start by downloading the free
GURPS Lite PDF from Warehouse 23. Try generating a couple of characters. Come back here and ask for advice if anything's not obvious. And then put the two characters you have designed through, say, something like the shootout in Marion's bar in
Raiders of the Lost Ark. Make allowances for the fact that everything will get smoother as you become more familiar. See how it feels.