How far South are you thinking?
Over to the South East you have Brands Hatch in Kent, whilst Snetterton is up a little bit in Norfolk. Across to the West a little you'll find yourself at Bedford Autodrome, but they have always been very strict on noise levels, so you might find yourself black flagged if you're not careful (Donington is also quite strict - I got black flagged for being 0.5db over their 98db limit). If you head up a bit further then you have Cadwell Park. Heading into the central area you've got Donington Park and Silverstone. Coming back down a bit you've got Castle Combe and Thruxton.
If you want a nice flat circuit to start with, try Bedford, Thruxton or Castle Combe. I've only done Castle Combe, of those 3, back in my old Nissan 200SX.
Silverstone has the history but it is incredibly wide in places, giving the impression of glacial speeds despite being over twice the national speed limit (I recall calculating Hangar straight as being something like 6 lanes of a motorway). Personally, it's the most disappointing circuit I drove, but you will probably get to exploit more of the power of the car than at some of the other circuits.
If it's your first trackday, try and find an evening that is specifically run for novices and spend a bit extra on a session with one of the circuit pros. Although there will be experienced trackers there too, the majority of people should be first-timers.
When I used to do trackdays in my 200SX, I'd arrive at the circuit, let the car cool down whilst in the briefing, adjust the tyre pressures to suit, check the fluid levels again (having checked before leaving the house) and then I'd head out on to the circuit and have fun. At the end of the day I'd wait for the car to cool down, reset the tyre pressures, check the fluids again and then drive home.
The only people that crash on trackdays are people that exceed their own limits. If you're out on the circuit and get concerned about the way somebody is driving, head back in and have a word with one of the marshalls. Definitely try and keep your sessions down to about 20 minutes max. (if it's an open-pit day) and let the car cool down for at least 15 minutes before going back out again.
I haven't been on track for about 10 years now, but I'm itching to take the F-Type out on some next year.
Hope that's useful.