Start by googling any random children's products brand, going to their website and seeing what stockists they have nearby.
Pick your nearest (ideally one that is not a chain) and visit their showroom. Try buggies out. Push them around, fold them up and down, put them into recline vs sitting up mode. Swap the carrycot out for the seat or car seat (they don't always have the car seat adapters on hand).
This will give you a list of brands and features that you like and dislike.
Some common things to look for/bear in mind:
Many brands are compatible with a range of car seats, others are only compatible with their own brand car seat. You might want to ask about this if the car seat is important to you.
Some pushchairs use a separate carrycot with mattress, some just have the seat itself lie down flat (no mattress), often with an optional "real" carrycot. Some the seat unit itself converts on the go (rare, but saves you storage space) and some you get one frame with two sets of fabric that zip onto the frame, making a carrycot or a seat. The last option saves storage again but is the least flexible since it's fiddly to convert.
For separate carrycots, some are suitable for overnight sleep (like a Moses basket) others are not certified as safe for this, but can be used for day naps under supervision.
Yes, if you're going to do off road regularly, you'll want big wheels, potentially air filled tyres.
Agree that there is almost always an official footmuff available but generic ones are also fine and tend to be much cheaper.
I don't think 3 vs 4 wheels has been a thing since about 2010. Don't rule out 3 wheelers, but no need to seek them out either.
You can keep it in the car but most people end up wheeling it into the house at least occasionally if DC is asleep. If it's damp or muddy it is best left unfolded so it won't go mouldy.
Less common things people don't often think of:
Look at how roomy the seat is for an older child. Some of them are really tiny.
The best buggy I ever had had machine washable seat fabric! This is rare but worth looking for.