Hi,
I don't tend to post on here (usually just lurk!) but as someone who has managed a fleet of coaches in the past for a large operator, please don't just type "coach hire Chester" into a search engine and go with whatever pops up. Especially if it's a "sponsored link".
It's easy to spot real coach companies as their websites will usually have images of their own vehicles with their logo or colours on the side. Many search results are generic sites with generic pictures and text (usually easy enough to spot) and are brokers. Please don't use these, and deal direct with the operator where possible. You'll know who and what you're getting and be able to speak to a real person. Brokers will just send your request on to any number of operators, good or bad, and add a fee for doing it. Most of the sponsored/generic sites are the same website for every town in the country, just with a different name, and have little or no local connection. The generic text and images are a huge give-away.
If you do want to use a broker, use someone well-known in the industry like CoachHireComparison.co.uk or NationalCoachNetwork.com. Both are 'real' and verifiable companies which can be trusted to supply a coach. CHC is owned by CMAC, which is ultimately part of ComfortDelGro, a major transport group. NCN is a Manchester-based company.
Better still is to look around for local coach operators (online, or even take note if you see a coach out on the road). I'm not local to that area, but having worked in the industry for a long time, some local-ish names with good reputations which come to mind are Anthony's Travel (Runcorn), Pat's Coaches (Wrexham) or Go Goodwyns (Clywd) (North Wales) but I stress they are just the first few names to come to mind, there will be others - I might be overlooking some equally good local names - and although I've every confidence that they are all reputable companies, obviously it's up to anyone to do their own research before booking.
In the vast majority of cases, there won't be a problem but if something does go wrong (coach doesn't turn up for example) then with a random "Coach hire Chester" broker, my guess is that you'd be left trying to contact them and hoping that someone would be able to sort it. With one of the reputable brokers above, I'd have confidence that either they or their selected operator would be on the ball to sort it. Dealing with a reputable operator direct, I'd be very certain of a vehicle turning up, and a replacement being sourced quickly in the event of a problem.
A Google image search for operator names will also help give you an idea of what their fleet is like (does it look smart, well maintained, etc) and help to reassure you that you're dealing with a real company which actually owns coaches.
In terms of vehicle sizes, 33-37 seats is about the top end of midicoach capacities (these days probably a Yutong TC9, Temsa MD9 or AOS Visigo). Bear in mind that luggage capacity is less on a midicoach than a full size one, and it might not have a toilet. A full size (12 to 13 metre) coach will generally be around 49 to 57 seats. If you want an idea of prices, you need to speak to the operator (or broker) and define what you need to get a number of quotes. They might vary quite a bit according to any number of factors, including how much they want the job, how last-minute the booking is, what vehicle specification you want, whether it's fitted with a toilet, the length of day (two drivers might be required), the exact schedule of where you need to be and when (are they just dropping you off and picking you up the following day, is the coach dropping you off, waiting, takin you to the hotel, taking you to the next day's location?) etc. As an operator, other than the maximum number of seats required, the fact that a couple of people will only go one way and a couple of others go the other way is fairly irrelevant as it's the group leader's responsibility not the coach operator's to make sure people are on board.
The more complicated the trip, the more I'd recommend speaking to an operator directly and being very clear with what you need exactly and what your proposed timings are, rather than asking on an internet forum.