For 分かる, you’d usually use が – the basic pattern is the ol’ ～ は ～ が sentence structure. を, according to my grammar dictionary, indicates “non-spontaneous comprehension”, such as when someone is trying to make someone else understand, or if the experiencer is making a conscious effort to understand. If you’re making a conscious effort to understand Japanese, then, you’d use を, whereas if you’re talking about English, which (I assume) you can understand just fine without thinking about it, you’d use が. At least, that’s my take on things.

For -たい, the choice of particle tends to indicate how much you want the thing – が indicates a higher level of desire than を. There are certain cases where you must use one or the other, though. If it’s going to be unclear what the direct object is, or the verb is passive, or you’re talking about someone else’s desire, you have to use を. For intransitive verbs, which don’t have a direct object, you need to use が.

And I wonder if those explanations just confused things further. =)

“Normally you would use the を particle with a verb” – don’t think of it as the “verb” particle, think of it as the “object” marker (or something similar). You’re right that it’s used along with verbs, but so are は and が and so forth. Generalising though, I’d say を is used for transitive verbs while が is the main particle for intransitive verbs. Compare「かれ を おこします」(I woke him up) and「かれ が おきます」(He woke up).