"I swam a lot today."
あまり goes with a negative verb (i.e. およぎません) and means "not much", "not very". Whereas, たくさん means "a lot" . So I think you got confused somewhere, because they are more like opposites than the same :p
Edit: Ah maybe I can see where the confusion comes from. Since あまり needs a negative verb, it is like "a lot" + negative verb, which results in " not a lot". But, "あまり+ negative verb" is like the opposite of たくさん.
Ok so exact answer is here which i have figured out<pre>
=>あまり : what u do usually, often or very much. ( Note : not talking about particular period. It's something u usually do) =>たくさん : anything u did a lot example : today i drank alot of milk, today ate alot or i will a lot today. I will buy alot of toys. ( Note: it's not something that u usually do. But in a particular manner u did alot)</pre>
So I just read in the comments of another "a lot" scenario question that たくさん is an adjective (used to modify nouns) and よく is an adverb (used to modify verbs). I used よく here, and it's wrong. :(
I would love further clarification on these two if anyone has more insight to share. :D
Also wondering about this... I am under the impression that generally たくさん roughly means "a lot" (referring to a quantity? usually objects?) and よくmeans "often" (referring to a frequency in time?). But that understanding doesn't hold up in phrases like よく出来た ("well done") and the above, where たくさん modifies a verb...
I'm still hoping for more insight on this, too, but based on a little digging online, it seems よくmeans something more like "often" or "frequently." たくさん seems closer to "a lot."
Since, "I swam often/frequently today," has a different meaning than, "I swam a lot today," that's my current best guess as to why よくdoesn't work here.