Translation:I swim every day.
it's also pronounced にち when used as a counter for days (excluding 2-10). 毎日 (まいにち) = every day. 一日 (いちにち) = 1 day. 2-10 take the irregular reading か instead of にち. after that, 十一日 (じゅういちにち) = eleven days, 十二日 (じゅうににち) = twelve days and so forth. 何日, "what day" or "how many days," is also pronounced なんにち.
"Everyday" (without a space) is an adjective meaning "happening or used every day" or "commonplace". It refers to a noun such as "clothes" that you might wear on a daily basis. 毎日 means "every day" (with a space), an adverbial phrase meaning "each day"; it's generally used to describe when the action in the sentence is occurring.
I go to swim every day... not right, oh, no. I go FOR A swim. Seriously, what's the difference?...
The difference is that going to swim means you are moving somewhere to swim, whereas およぎます just refers to swimming, without saying anything about going somewhere to do so. I imagine your sentence would have 行く in it somewhere.
Considering that logic, both are equally correct. The "verb+to+infinitive" is used for telling the destination of a verb, while the "verb+for+noun" is used for telling that the destination is a noun-ish thing (a drink, or a swim, etc.). What you refer to is the difference between the gerund and the to+infinitive -going swimming and going to swim.
You said correctly though: with およぎます the verb is the swimming, while if I want to say: go to a swim, it should be something like およってだいきます, or something like that. I'm still bad with the te form-type verb+verb structure...