"I swim a lot in the summer."
You use ni to indicate a specific period of time. Wa is more general. It translates more like during (the period of) Summer it's more relaxed and general. Ni is more specific, so it translates more like IN Summer (specifically) - you'd use ni to indicate that you are specifically swimming a lot at Summertime. Duolingo didn't convey the difference between wa and ni very well at all.
It is the case that よく means "often" and たくさん means "a lot" or "many", but it seems like たくさん can be used in place of よく when talking about the frequency of something. Whichever you use, you would definitely be understood in that situation. The reverse is not true, however. You can't use よく in place of たくさん when it is being used to describe quantity.
Pretty sure it should be に instead of は because summer is a period of time. The whole sentence should be 私は夏にたくさんおよぎます。With I (私) being the subject and so therefore marked with は.
I believe you can also combine the markers to make 夏 には たくさん泳ぎます。Which in English is essentially the same sentence (in the summer I swim a lot).