Translation:I swam a lot today.
たくさん = many/much/a lot, and comes with a positive verb. E.g. たくさんの本を読みます -> "I read a lot of books".
あまり = often used as 'not very (many)'/hardly, and comes with a negative verb. E.g. あまり本を読みません -> "I don't read many books"
Note: あまり can be used with positive verbs too, in which case it means "too much" (or, when written as 余り; "the surplus/remainder/rest"), but you'll probably see that less frequently.
You can't really tell, you just have to learn and remember the cases in which that happens. Most of the time, when you see a word with multiple kanji, you can combine their individual pronunciations (the Sino-Japanese ones, aka "onyomi"), but this is an exception. I suspect that's why Duo shows the kanji sooner rather than later.
Technically speaking, this could be either, as well as (s)he, we, or they, since Japanese verbs do not change with the speaker and hardly use personal pronouns in general. However, as Ryan says, Duo seems to have used "I" almost exclusively throughout its Japanese course. (unrelated to verb endings though!)
Hi Ryan, this is probably not the answer you want, but I have noticed so far DL is presenting only sentances that are statements by first person, so I am guessing "imas, imasen, imashita", etc. endings all have I or we as the speaker. I am sure someone else can give both of us a better explanation. : )