Classification of anything in linguistics always has to be focused on what you want to achieve. All classifications come from a certain perspective. So there will never be a definitive answer.
Having said that, it may be useful to have a bit of a point of reference when reviewing the literature.
You must also not confuse classification based on properties intrinsic to the verb or its use with the classification of the actual uses. So e.g. infinitive / finite verb describes the state of the verb, as does the person or tense. Transitive, on the other hand, is an intrinsic property of the verb (or at least one of its senses).
You'll find people talk about different classes of verbs in two broad categories: syntactic and semantic (they overlap and interact but it's useful to keep the distinction in mind - no matter how provisionally).
- Transitive/intransitive/ditransitive (to see sth, to sleep, to give sth to sb)
- Main/auxiliary (I have a book vs. I have seen a book)
- Modal verbs like can, must, may, should, etc. (although the primary definition is semantic, what sets them apart are syntactic properties)
- Phrasal verbs (walk in, put up - sometimes called 'prepositional' even if not all particles are prepositions)
- Irregular verbs (this is more of a morphological class of verbs with irregular participles e.g. go - went - gone)
You will find these (and more) in most grammar books.
- State verbs (to know - has syntactic implications because will resist -ing as in I'm knowing) v Action verbs (to kick) is probably the most common axis of classification with many variations and elaborations
- Perception verbs
- Verbs of motion
There are virtually no limits to semantic classifications of verbs. They are also less likely to be useful for learners. You will find them in research literature rather than use guides and grammars.
Some of these categories will be exclusive, others complementary. So for instance, a single verb cannot be transitive and intransitive at the same time. But a verb can have multiple senses, so for instance 'I see' vs. 'I see you'. Some categories do not apply to some types of verbs. So for instance, the transitive/intransitive distinction does not apply to modal or auxiliary verbs. Some semantic categories will tend towards certain syntactic categories but with many exceptions, so verbs of motion will tend to be intransitive (to go, to walk - but to drive sth).
Of course, then there are also cross-linguistic comparisons related to verbs. For examples, some languages tend to express the manner of motion as part of the verb (verb-framed, e.g. enter) and others do it with an external particle (satellite-framed, e.g. come in - more common in English).
Finally, remember that few of these classifications will work across languages.
For instance, German has strong/weak verbs, Russian perfective/imperfective, etc. Modal verbs in some languages don't stand out as much as they do in English. Some languages don't have the infinitive, etc. Many languages will classify verbs according to their morphological properties into various classes and models sometimes numbering in the low tens.