The weather in March is generally warm (80's) but still can get occasional colder spells. It's often quite windy, but you shouldn't see much rain as a general rule. This year Summer was quite dry but we've had a lot of rain in the last couple of months (when it would normally be much drier), so who knows what's going on !
There are a number of walking trails on Big Pine. A good starting place would be the Key Deer refuge center who can give you details, maps of the trails etc. they likely have them on their website too.
They have recently improved the bike path along U.S.-1 significantly, so cycling around is a lot more attractive than it once was. It's almost all sepparate from the highway now, so much safer. Theres miles of quiet streets around Big Pine also, way back off US-1. There's a bike shop on Big Pine that I'm pretty sure rents bikes. If you plan on using a bike a lot you might be better to buy a $150 bike in KMart in Marathon and sell/give it away when you leave.
There's also good kayaking to be had. I've put mine in at the end of No Name Key and explored that area before, but there's loads of other places if that appeals to you. KMart also sells some inexpensive kayaks (<$200) which are OK to paddle shorter distances, as renting them can be expensive if you do it regularly.
I'm honestly not KMarts biggest promoter, but some of their cheap stuff makes a lot of sense to buy as you're on a longer trip. If you were here for a week I'd say support the local businesses and rent things like that when you need them, but for a month it might be more viable to buy.
You should definitely visit Bahia Honda State Park also. I think you can take dogs in there, but best to check as I'm not sure. It's beautiful, and only something like 8 or 10 miles North of Big Pine.