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.
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...