English prepositions can do two jobs: 1) changing the noun like "on the floor", and 2) changing the verb like "to get out". Unlike English ones, Russian prepositions do only 1). For 2) we use verb prefixes exclusively.
This concept presents in many languages and even in English you sometimes say "bypass" or "foresee" etc. What makes English special is that in 80% cases you still may use separate prepositions like "go in, go out, go on, go by" so there's one monstrously large vocabulary entry on "to go" yet it's only one and you know what letter it starts with. But in a Russian vocabulary there are a dozen of entries on those "to go"-variants and they all start (the prefixes!) with different letters.
How to deal with it? Well, think of it like an extra set of "prepositions" which you have to learn. In fact, they sometimes coincide. Let's take an example.
"Под" means "Under" yet it's a verb prefix too. Let's add it to a verb "Держать" (to keep, to hold). Now we have "Поддержать" (note that double D: there's also both preposition and prefix "по", so "подержать" is an absolutely different word). What "Поддержать" means? Literally "to hold (from) under". Then applying some mind work gives "to support" which is the right answer.
The problem is that the prepositions (both true and "prefix-like" ones) have too many meanings so you can't deal with only a simple one-to-one correspondence. Or at least it wouldn't help you to get it. As an example, "Подделать" is "Under" + "Do". But what it means? Really, "To Forge". Having known that beforehand, perhaps, I can speculate that it was due to forging signatures or stamps which are under the text, yet no chance I could figure this just without any context. Sadly, this means you still need the dictionary.
But what about listing all those prefixes? Well, I don't know the comprehensive source to provide it here. But there shouldn't be too many of them (or else Russian people can't remember them all). So let's try to deal with it just by another example. How many verbs can we form out of "Держать" by adding different prefixes? Now let's do it: starting from the literal meaning we go to true one.
Выдержать ---> "to hold from inside to outside" ---> to withstand, to survive, to keep self-control, to make vintage wine or cheese etc.
Додержать ---> to keep upto/until smth. (rarely used in prefix-form)
Задержать ---> "to hold after" ---> to delay
Издержать ---> "to hold from" ---> "to hold completely" ---> to spend, to waste (a little bookish)
Одержать [победу] ---> "to hold [a victory] all round" ---> to win
Передержать ---> "to hold throughout" ---> to hold too much
Придержать ---> "to hold near" ---> to delay a little
Подержать ---> to keep a little
Поддержать ---> "to hold under" ---> to support
Сдержать ---> "to hold from top down to bottom" ---> to hold back, to fight off
Удержать ---> "to fulfill the process of holding" ---> to keep despite of smth.
This list is no way complete but it could serve as a starting point. Try to apply these same prefixes, say, to verb "Идти" (to go) - can you now understand all the verbs? Выйти, дойти, зайти, изойти, обойти, перейти, прийти, пойти, подойти, сойти, уйти? (The life is hard and making the couple of errors here seems to be unavoidable, yet there always is the vocabulary to fix them).
UPD. As I was pointed out any "preposition" matching only the verb is considered to be an adverb not a preposition anymore. That makes much sense and yet raises one more question: if Russian language has adverbs too (and it really has) why Russian verbs need prefixes at all? The answer is productivity. Naturally, some adverbs make a bigger meaning shift than others. Say, "Go forward" still means "Go" plus a direction. While "Go nuts" isn't about "going" at all. So sometimes it makes sense (at least for Russians) to merge two words into one.
Let's take another example. "Я повернул налево" (I turned left). Here "налево" (left) is a regular adverb. Yet the verb "Повернуть" is still uses a prefix: "По" + "Вернуть". That is "an incomplete action" + "one turnover" which gives just one regular turn in a grand total. It's fun that in English there's a prefix too although the other way round: "To Return" (Вернуть in Russian) is formed like Re + Turn that is turning twice gives coming back.
Thus there's nothing truly new in the concept of prefixes, and once you've got it, it helps much just like common prepositions, adverbs and conjunctions do. Though the flexibility is always a danger. Not all the couplings really make sense and combining them blindly is a way to play a fool. The most well-known joke about prefixed verbs is the word "Недоперепить". Could anyone translate it? :-)