Don't worry, there are lots of options and you'll work it out :)
Any time your baby is in a car they need to be in a car seat. Legally, there is an exception for taxis, but I would not recommend using this exception unless you have no choice. The car seat is basically your child's seatbelt if you were to crash. A sling is not a safe option in a taxi.
Car seats aren't usually totally flat, and it is best for babies to have time each day lying flat so that they can develop the muscles they will need for rolling, sitting, crawling etc. It's not a big problem that car seats aren't flat, because you won't be in a car for most of the day anyway. They can have time lying flat at home, in their cot, in a flat carrycot/bassinet in a pram, on a playmat or blanket on the floor, etc.
You can use most infant carry type car seats on a pushchair, and it's OK to do this sometimes for convenience, but if you're planning to use the pushchair a lot, it's worth getting one with a flat lying position or a carrycot/bassinet attachment (carrycot and bassinet mean the same but carrycot is more British English whereas bassinet is more American - they're interchangeable.)
There is a further issue with car seats in that they are angled at around 45 degrees when they are in a car. This is important for crash safety, but babies kept in this position for too long can occasionally struggle to get enough oxygen. Because of this, it's important not to keep babies in car seats for too long at one time. For example, if you put them into the seat at home, then drive to the shopping centre, move their car seat to the buggy, walk around, go back into the car, the baby falls asleep and you carry them asleep into the house in their seat - this may add up to several hours in the seat without a break, which is a problem. The recommended max time in a car seat used to be around 2 hours but there is now no specific time limit recommended and the guidance just says to be aware of the risk and give babies regular breaks from the car seat.
IME, this is fairly natural anyway especially if you take them out of the seat to feed them. One of the benefits of the carry type seats is that if the baby falls asleep you can move them in the seat without disturbing them, so just use common sense here - if they have only just fallen asleep, they will probably transfer to a carrycot fairly easily if you start walking straight away. If they are likely to wake up in the next ~10 minutes, personally I'd let them finish the nap in the seat.
The risk is higher for newborn babies, as they don't have the muscle control to maintain their own airways and don't often get a perfect fit in car seats anyway. The guidance used to be 30 minutes at one time during the first 4 weeks. That one can be a bit more difficult to stick to, which is why the current guidance is just to limit car travel as much as possible for newborns.
To use taxis like you say I would probably use the car seat on the pushchair chassis - use the car seat strapped into the taxi, then use the car seat as your pushchair seat when in town, making sure to take baby out every so often to give them a break (which they will want anyway).
Or you may find a local taxi firm who can offer a car seat for you to use. Or you may meet another parent who can drive and will pick you up so you could leave your car seat in their car for the day.
Or you may find a bus route that takes you into town- that was what I did most often. The safest place for a baby on a bus is in a pushchair, ideally in the wheelchair space (if a wheelchair user doesn't need it, of course) facing against the direction of travel with brakes on.
If either your town or your families have black cabs then you can also wheel a pushchair directly in here with the same principle but bear in mind this is likely not as safe as a proper car seat. It's not so bad at low speeds in city traffic, but I wouldn't really want to do this in a town travelling 30mph+.
To visit your mum I'd probably take the train rather than a coach, because you can't usually fit a baby's car seat on a coach, and then either keep a car seat at her house, again find a taxi firm with car seats, or bring the car seat on the buggy.
Bear in mind that to do any of the car seat/buggy combos you're looking just at the first stage of car seat which usually lasts up to approx 1 year. Some of them do last longer - Cybex Cloud might be worth a look, especially since it lies a bit flatter, so would be more comfortable as a buggy seat. Babyzen Yoyo for your pram perhaps. But once your baby is bigger you'll still be stuck because they still need car seats and they can't go into the more portable booster seats until ~5/6 years old minimum.
If you're not pregnant yet and you don't want to learn to drive, would it make sense to look into potentially moving closer to town so that you can walk everywhere? When I was looking at places to rent with a toddler on my own I had accessibility (walking access to things I needed plus bus routes) as an essential criteria. If it didn't have those things I didn't even look at it.