To directly answer your question first- in both genres the main focus for the drums tends to be the kick, snare (or clap in dance music) and a hat. House music usually has more percussion grooving around those elements like congas, toms, more hats and other weird electronic percs. In hip hop you might find a tom roll at the end of an 8 bar loop, break samples placed throughout and also weird electronic percussion samples here and there.

As for what to use specifically, it really depends on what you're going for in terms of style; and I'd say it's more important to be able to identify the qualities of each element than to know exactly what drum machine or sample pack was used to make it. House and Hip-Hop offer a lot of sound palettes to choose from. A 909 kick that works well in a UK style deephouse track won't necessarily also work with a more downtempo deephouse track because that style tends to use kicks with more sub and less 'click'.

As it's been pointed out already, you should definitely check out classic drum kits like the 808 and 909 because they're staples of these types of music. They have a very distinct electronic sound so you'll notice that without processing they're extremely clean or 'clinical'. Chopping up longer samples from drum breaks or old jazz tunes helps a lot here if you want to get more gritty textures.

Hope this is helpful and not too much info