We didn't get a car until DD was 3, and I didn't learn to drive until she was in school. We live in a village, but there was a bus into town every 20 minutes, so we had relatively good public transport. It's absolutely fine.
Tips:
Choose your pram - we went for one with plenty of luggage space underneath for doing the shopping. The only snag was when we discovered it wouldn't fit in my parents' boot, but we got round that. It wasn't a good pram for anyone who wanted to put it in a car, but it was great for us.
BabiesRUs were fantastic at advising us on the car seat which fits the largest number of cars - it was useful to have one so that we could accept lifts, but of course it was no use if it would fit the car. We were before isofix became common, so I don't know how that will affect things.
Although it's always best to have proper car restraints, there are two exceptions to the law: taxis, and short local journeys in unexpected circumstances. I think the latter boils down to "your friend won't be fined for giving you a lift to A&E". ROSPA advised us that if we had the pram, we should ask for a wheelchair accessible taxi, and put the pushchair facing perpendicular to the direction of motion. The other thing we did was to get me in and belted, and then put her in the sling (when small) or put the long bit of her reins around me, so that she was secured to me.
Reins can be hooked round the back of some bus seats to help keep a toddler anchored.
If you have a longer bus journey, make sure a toddler has been to the playground before you get on the bus home (or go to the library with a crawler) so that they aren't restless. (That applies to long car journeys too.)
Holidays: we usually holidayed in this country. We had a jumbo suitcase which held enough for all three of us for a week (buying nappies on arrival). Many holiday lets can supply travel cot/highchair. Having a half hour train connection was not a bad thing - she could run off a bit of energy in the station.
We did A&E on the bus twice, for injuries. It turned out to work really well - she fell asleep in the pushchair and was then rested and cooperative when we actually got to see someone.
Some positives:
You get to have so much more communication with your baby/toddler sitting next to their buggy on the bus than you do when you're driving and they're rear-facing in the back of the car.
You can feed them on the bus (when my parents gave us a lift to a family event, it was a bit of a shock to me that we had to be up early enough to feed her before we left!).
Mine had all her naps in the pram (pushchair later) - often falling asleep on the way home from baby groups and then stayed asleep when we got home. That might work less well for you if you have stairs to your flat, though.