Both are correct. :)

There are verbs that must be followed by other verbs in the infinitive (using TO), and others must be accompanied by verbs in the gerund (using ING). And after modal verbs you never use the infinitive verb with TO ... correct?

Both are correct. Blair is right that "to" is not necessary each time

Looks fine to me. You don't have to put 'to' before each subsequent verb.

The first one is done correctly! The second one requires a little change, so it becomes “I like to run, to walk, and to swim...” Antonio, I hope it helps, :)