How can I learn to individuate ancient Greek verbs?

No substitute for rote, I’m afraid. But there are patterns and regularities, and you’ll need to make them your friend:

• If anything looks like a preverb (prepositional prefix), strip it off. It’s usually a safe bet that it is in fact a preverb.
• The endings do have patterns (the final vowels/consonants, the thematic vowels, the verb stem endings). The more comprehensive grammars present the patterns as part of their historical approach, but they aren’t just of historical interest: they help you learn the inflections. (And native speakers use these patterns subconsciously.)
• The thematic vowels in particular are your friend, because they help you get to the present form: whether it’s -αω -εω -οω , or  just -ω.
• If it’s athematic, well, that’s a matter of rote memorisation. Good thing there aren’t that many of them, and they tend to be common. Which is after all the story with irregular verbs in language.
• Be on the lookout for second aorists and liquid verbs, they are an added complication
• Think in terms of proto-Greek: it will help the patterns become more obvious. So don’t just learn the tables, but where the inflections came from. Uncontracted verbs, after all, are mostly proto-Greek, and certainly not Attic. The tendency  to drop intervocalic /s/, for example will make 2nd person passives look much more plausible.
• Dialect is indeed harder. It took me three hours to work out what φαῖο means (The tale of φαῖο), and it didn’t help that the form, being dialectal and in a new edition, was not in the standard grammars.
• Byzantine Greek is even worse, because there is a *lot* of fantasy morphology. Much like the Irish monks, Byzantine writers often just made tenses up.
• In the olden days, there were dictionaries of verbs. Lexikon ueber die formen der griechischen verba, Traut, Georg  is one instance; and Greek verbs, irregular and defective, Veitch, William is another. Veitch is comprehensive; the advantage of Traut is that you can actually look up individual inflected verbs in the canonical corpus. (Just remember to strip off the preverbs.)
• There are morphological analysers of Greek, which are hyperlinked in to some online texts. Including Perseus, and the TLG (both the subscription full version, and the free subset). The TLG’s morphological analyser is an ongoing project that I’m working on, and it provides the possible analyses of word forms in texts.