Time trialling can be great fun.
The CTT website lists clubs across England & Wales that are affiliated to it. Have a look & get in contact with clubs in your local area. You can sound them out. The 'season' runs from March to October and during that time most clubs have a mid-week club event (usually 10 miles, but there's likely to be some 25s and at start & end of season the odd 5-mile event) where most will be happy to have you turn up and give a TT a go. Expect to be asked to make a decision on joining the club after you've tried a couple of events.
Once you've joined a club affiliated to CTT, you're able to enter 'open' events run by other clubs and which form part of the annual calendar of events. There are events pretty much every weekend during the season, with mid-week open events too. The term 'open' means 'open to members of CTT affilated clubs' and are usually advance entry. It used to be by posting off an entry form to the organiser, but there has been steps made in online advance entry. Don't know the current situation on that as the offspring is no longer cycling competitively. The reason for advance entry was due to limitations on the number of riders allowed on any given course, and field layout in terms of rider start time and expected finishing time to make life practical from a timekeeper's POV. Open events have monetary prizes. These are usually 'scratch' = fastest overall & equates to adults under the age of 40, with 1st, 2nd & 3rd places, Juvenile & Junior prizes if there's youngsters taking part. For some reason they seemed to ignore espoirs category, even though it is an offical age category and has been for many years..Then there's the various veteran categories (a vet is anyone over 40, and the vet standards are available at CTT web site), so there's prizes for vets, then there's the handicap prizes, which are aimed at those new to the sport usually, the slower riders, to give them something to aim for.
So you'll see that there's a lot of awards on offer at most events - usually cash enough to cover entry fee - not huge - riders aren't in it to make a living, but to enjoy the sport.
You can do TT-ing at many levels, from "I'm slow, but I enjoy riding and don't care that I'm slow" to "last time I did this distance/course I did X time and I want to see if I can beat it" to "I am lord of the universe and you can't catch me na-na-nana-na!"
As for courses, there's all sorts. From dual carriageways to country lanes and everything inbetween. Course distances are formally measured, and risk assessed - traffic levels are taken into account and if over a specific level, to course is not used. Similarly weather is taken into account and if the conditins on the day would mean the course is too dangerous to ride from the organiser's POV, the event will be called off. Riders are expected to adhere to the HC & ride safely - ride with your head down, not watching where you're going and you may well find yourself disqualified, and drafting another rider is a definite no-no leading to disqualification (riders are set off at one minute intervals).
My son thoroughly enjoyed time trialling for several years. Now he's at university, he's not competing as his studies are coming first (fair enough).