I am a bit unclear as to whether "tre" or multe" should be used with particular verbs. For example, should I say "Mi tre ŝatas ĝin" or "Mi multe ŝatas ĝin"? Similarly for ami, voli, and deziri. Are there cases where multe and tre can overlap in usage or are they separate?

For me, in these examples, you can use both with exactly the same meaning.

If you write "multe" on http://www.reta-vortaro.de/revo/ you will read that they talk about that.

karibdo:Are there cases where multe and tre can overlap in usage or are they separate?

There's more overlap than in English. In English one wouldn't use "very" with a verb, but in Esperanto one can do so. We wouldn't say say "I very like it", or "I very laughed", but in Esperanto you could say Mi tre ŝatas ĝin or Mi tre ridis.

However, I would stick to translating "very" by tre and "much" by multe, unless you remember examples of the overlap.

The distinction can be thought of like this:

Tre: in a high degree, or with a strong intensity.

Multe: in large quantity, with long duration, or often repeated.

They are not generally interchangeable. See here in PMEG for a good explantion and examples:

http://bertilow.com/pmeg/gramatiko/kvantaj_vorto...

"multe" is probably the safest to use with verbs when you can't tell what you should use, but in these examples "tre" will still make sense, but since "tre" is mostly used with adjectives, sometimes it doesn't work well for verbs, if at all.

I doubt you could say "Mi tre kuras" easily, where as "Mi (tre) multe kuras" is easier to figure out.

Also, -eg- works well too. Mi ŝategas, mi amegas, mi kuregas, etc.

Eg is often used with both adjectives and verbs fairly interchangeably, but the meaning can sometimes be different to what's expected. But you've got a lot of expressive power, just get familiar with what words and suffixes do what and you'll start to be able to make on-the-fly judgement calls about these things.

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EDIT: Maybe here's a rule of thumb - if you can turn your phrase into a passive sentence (so rather than "mi ŝatas ŝin", you have "ŝi estas ŝatata de mi") and the "tre" still works in front of the passive adjective, then it should make sense as a verb.

"Ŝi estas tre ŝatata de mi" = "Mi tre ŝatas ŝin"

BUT, since "Pomoj estas tre manĝata de homoj" doesn't really make sense, you can't say "Homoj tre manĝas pomojn", you can only say "multe" or "tre multe"*.

*(in this case, "ofte" would be even better IMHO, but that's just me being pedantic about style)