You don't need a car. We didn't have one when ds was born, we only got one when dh needed one to commute to work as public transport would have taken hours and hours. Of course since he had the car to work I was a SAHM with no car. It was fine.
There are some frustrations, for example you still need car seats for when you get lifts and they take up a lot of space in your house!
Whenyou get the train somewhere you need to either have decent public transport at the other end to get to your destination or have someone pick you up which means you need to bring the carseat on the train too in a ddition to the buggy and so on. However, unless you visit people in other cities on a very regular basis this isn't an issue too often. Long train journies can be annoying though, in a car baby tends to fall asleep and stay like that untilt he car stops, on a train it can be a longer trip and changes can be hard when you have a baby, suitcase, carseat and pushchair to unload, shove tothe other side of the station and reload.
Initially, especially in an area with decent buses you will have plenty of baby groups and toddler groups in walking distance or a bus ride away. However, as your child gets older and you want to do things like soft play (especially in winter!) or farms or just bigger parks, country parks and such, they tend to be out of town and often not very easy to get to on public transport so you either can't go or have to rely on lifts. Lifts are fine when you and your friend have one child each but as soon as you start adding more then you run out of space in the car.
Mostly though it is fine, if you manage now then you'll manage with a baby. Just look into decent slings rather than a pushchair all the time as they are easier on buses and when buying a pushchair get something that folds down easily as you can't guarantee space on the bus, there may already be another buggy or a wheelchair user may need the space. You will either waste a lot of time waiting for the next bus or need to buy something practical that can fold and be prepared to do a lot of juggling of shopping bags and baby.
Start online shopping for the bulk of your groceries (we started doing a monthly shop of tins and dried goods, household stuff etc) so you only have to get your fresh food, saves on getting to the checkout and finding oyu can't actually fit all your shopping on the pushchair!