When: I would say between Mai and September. However you might be even lucky in April. I was there at that time and even had a swim in the ocean (although the water was still very cold).
Get there: When you are in Lagos you can choose the beaches on both sides from the city. I would recommend going right (facing the ocean). There is a little walkway on the coast that gets you to so many different beautiful beaches. You can get maps at most hostels and hotels.
Do: Walk everywhere. I don’t even know if all beaches are accessible with car (some definitely are). I really enjoyed walking from beach to beach discovering different routes to get there. At some point (if you walk far enough) the ocean changes from Mediterranean looking to more of an Atlantic style (I actually don’t think it is the Atlantic yet but the two oceans meet at some point close by). There you can take a surf lesson if you like since the waves are bigger.
My tip: Bring lots of water and maybe some food with you. There are beach bars at some of the beaches but not all of them and you will get hot during walking. Also wear sunscreen and a hat since you will be exposed to sunlight for most of the time.