Depending on the context, あ and う could either be pronounced as one vowel after another with a clear but subtle space between them or as a diphthong. English has tons of diphthongs too. Take "night" for example. The "i" in night is NOT one sound. Phonetically, it's N-AI-T, right? The "i" is A + I in quick succession. The English word "cow" has a very similar diphthong to あう. The "ow" in cow, isn't pronounced like "oh" how it looks, it's pronounced like AU, Ah + Ooh, Ahooh, or AU. [tatakau is pretty much ta.ta.cow] Starting with "ow" from "cow" is great, because it approximates あう, but you want the EXACT pronunciation right?
Well, turns out the English language really has neither あ or う in it! The Japanese あ "a" sound is very short. It's precise phonetic location is between the "ah" sound in Canadian/General American "pot" and the "a" sound in central Canadian "cat", NOT American "cat". American "a" from "cat" is usually a slight diphthong, so don't even compare あ to the American cat sound. If you know Spanish or French, the short "a" from those languages is nearly identical with Japanese "a". Let's try it! Hold the "a" from "mat" in a Canadian or Californian accent, now drop your jaw ever so slightly, you should be bang on the Japanese "a". Now you can say "Matte"!
Now for う! Dun, dun, dun! Whenever I hear native English speakers trying to pronounce this I usually cringe, but it's definitely possible to master it. I have. There is nothing close to the Japanese "u" in English, not even similar, phonetically speaking, so this'll be a learning curve if you're a beginner.
Americans tend to, again, diphthongize their vowels. "Ooh" is no exception. Whereas Canadians do so less often. If you watch cartoons/dubbed anime, many voice actors are Canadian, so shoot for cartoon character phonemes LOL. Say, "boo!" but don't make the "ooh" like "iuww", make it a pure "oo" sound like Arnold Schwarzenegger would say. See how low and dark that vowel sound is? It's in the right ball park, but much too dark and rounded right now.
Now say "ee" and without changing the position of your tongue, round your lips into an "oo" shape. The sound you're now making should be like the "ue" sound from German, like in "Uebermensch". I can't use the umlaut, so I'm using the digraph "ue" instead, same diff though. Now, drop the tip of your tongue in your mouth ever so slightly, the sound should be less "squished" sounding. Now, unround your lips, make your lips "flat". The sound you're now making should be the Japanese "u" or thereabout. Make it "cutely" nasal, and it'll be even closer.
The sound should be more squashed than our "oo" sound in English, but cuter than the "ue" of "ueber". The back of the throat should be constricted like our "oo" as in "do", but unrounded like our "oo" from "foot". In fact, thinking of it as somewhere between "oo" from shoe and "oo" from "foot" might help you. Just make it a bit nasal and squished with your tongue nearer the palate. You can listen to the right sound in any anime or on the wikipedia article for Japanese phonetics. The IPA vowel chart doesn't have the exact one because it's unique, but a native Japanese girl recorded it for the Japanese phonetics article in the vowel section.
Hope this helped! Oh, and remember, make your diphthongs tight like in English. In tatakau, instead of going ta.ta.ka.oo, go ta.ta.cow - like 1 syllable at the end there.