R.E.
One great thing i've found about Tallahassee is the multitude of parks- Lake Ella is my son's favorite, because I put him in the stroller and we walk round the lake and watch the ducks.
When she's a few months older (crawling at least), there's a great place in Market Square (Off Timberlane) called Baby Power, which has baby play on Monday and Friday afternoons and Saturday mornings for $5 a pop.