Track days are normally run in a 'open pit lane' format which means you pay your money and can go on and off the track as you please. There are only a specific amount of cars (20 - 30 at a guess) allowed out at any given time.
3 of the main track day organisers I can think of are Javelin, Opentrack & MSV. Check out their sites to see info on how the days work and what days the have available. There are barely any track days currently that are not sold out as everyone is getting them in after the COVID-19 situation. Donington will be closest for you I imagine, its a fairly safe track with plenty of run off if you do make a mistake, which you wont if you drive to your limits. Days can cost anywhere from £150 for a full day to £300 dependant on time of year or even more if it's a special track day. Donington also offer evening sessions at around £150 currently.
There is no insurance out on track unless you take it out with your insurance provider (if they do track insurance). Just drive steady and to your ability and you wont need it anyway.
One of the main things you will need is good brake pads (Ferodo DS1.11, Pagid RS29 etc) & fresh good brake fluid. Also road tyres will soon overheat and turn to mush on dry days so it's better to go with a more track oriented semi slick.
Just get out and try an evening session imo and see how you like it. I got the track day bug a few years back and love them.
Also dont expect to go out and be the fastest as some people sink over £20k - £100k into their track cars and some are semi-pro, professional drivers. One of the things that shocked me when I did my 1st day was how slow my driving was.