Focus! What is it exactly you want to accomplish, and how? Tampa to the Keys, and back, in a "few" weeks is insanely reasonable on your boat. What I would suggest is to sail from Tampa toward the Keys and see where that idea takes you. I have learned from our Gulf coast that there is much more along the way than any imagined destination can yield. As Benny pointed out you have a limited number of hours of daylight per day this time of year (figure 6-8 in reality along that coast this time of year). Figure an hour getting to and from open water, and never consider reaching unknown anchorage after dark... If you enjoy a beer you will lose sailing time. Now, on my H23 I've learned if you have a crew they will mutiny in less than 7 days. If you sail alone 14 days will put a funk on you Ajax won't scrub off. Three weeks will leave your back and knees to the surgeons.
You have plenty of good anchorages from Tampa to Ft. Meyers, and then you have to go offshore and have few options. Choices become limited below Sanibel with my 3' draft.
New Pass is great anchorage, but requires a careful approach from the SW for the shoal reaches out quite a way. from there sail a day and make Marco, anchoring in Coconut Bay. Take the Marco River to save yourself the grief of the Romano shoals for a short sail to Everglades City, shooting for Indian Key amongst the Ten Thousand Islands. Even if you choose to anchor in Russel Bay I'd recommend the trip up the Barron River just for the sight of the channel along the way. I haven't been able to pass this destination (since running away the first time) without spending at least two nights for I enjoy the ambiance of this town's ecclectic history, but at first it seems scary. The pirates are more clever now, and quite friendly. Don't be fooled.
You can reasonably consider skipping Everglades to make Shark River and save a day, or two...Shark River offers fantastic anchorage. Again this takes an approach from the SW as do most anchorages along the Gulf along raw shore. I've sailed from there through Whitewater Bay to Coot Bay down Buttonwood Channel then over the "plug", but that was prior to Hurricane Charlie. I doubt that area has recovered so the lift over the plug probably doesn't work, still. Even if it does you have to step your mast to get there, and then if there is a nest on it they won't operate it for you. Cell phones don't work well north of Midway Keys inshore. Screw it; just stay offshore after Shark River. On a good day you can make the Keys from there, or consider making an overnight at Flamingo and going from there.
From there are the Keys you desire, and the choices abound. Key West is three days. Largo is a day. Biscayne Bay is two hard days to reach it's north end. If it is the Keys you want do as others suggest and just drive down there in a few hours. Focus, not overlooking what is directly in front of you.
