Depending on the speaker, these spellings represent either two or three sounds. The short answer is that for standard pronunciation, you can treat the vowel sound in ta and tan the same way, but the a in tang is further back.
The a in tang is a back open vowel, written [ɑ] in IPA (the International Phonetic Alphabet). That means that it's the same as the normal a in ta, except further back. The difference is not very big, but shouldn't be hard to hear. This sound is close to the a in Received Pronunciation, as in words like bath. This means that the word bang in English [bæŋ] is not very close to bang in Chinese [pɑŋ].
The -a in tan is trickier. There is some variation in how this is pronounced, varying from a normal [a] (i.e. the same as in ta) to [æ], which is less open, similar to the vowel sound in English bad [bæd]. The [a] pronunciation is more standard, but it's common to hear sounds approaching [æ] as well. You can find both transcriptions depending on where you look, so either [tʰan] or [tʰæn].
If you listen to the top row (-a, -an and -ang) on the Chinese Pronunciation Wiki, you can clearly hear the three different pronunciations (including something between [æ] and [a] for -an).
So, to answer your question, you can pronounce the -a in ta and tan more or less the same, but you should make sure that the -a in tang is further back.
The same is true for the other initials, of course, not just for t-. It's true that the initial (or any sound) influences the final (or any other sound), but I've never seen anyone claiming there's a significant difference between similar initials that learners need to worry about. These influences are often the result of transitioning from one sound to another and will happen naturally. You can certainly treat the -a in ta, da, ba, pa, ma, na, etc. the same.