DD was only a few weeks old - maybe 3? I hadn't got my own car yet (was driving years but DH and I had always done fine sharing a car) and I used to go down to the "town" walking the pram, have a walk then a coffee in a coffee shop (and give DD a feed), then we'd get the bus back up the hill home. Sometimes I'd even get the bus both ways if it was rainy, to get out into some fresh air and other people while DH was at work. Never got off just because she was crying (between reflux and other issues - we'd a lot of crying!)!
Train was later, just because of logistics (it didn't go where I was interested when DD was very small). And I got my own car when DD was 7 weeks old so we were more independent again.
She was 8 months old for her first flight, I kept up bf until shortly after holiday that so was able to feed her for takeoff etc.
We used to get the bus a LOT for commuting when she was young - I drove her home (DH having driven her in) mostly up to about 2.5 years. DH started cycling with her on the back when she was about 16 months old (could hold herself up well, and warmer summer weather - but she went in on his bike all through winter up to starting school (he got knocked off and broke his wrist on his 3rd last day of doing that commute - which suddenly became his last). But when I moved building, and DD moved building near my new one, when she was 2.5 years, we tended to commute home by bus or occasionally train.
Generally no issues at all. Except in very early days when she was a very small baby (so trickier to manage and fold proper pram alone) and I wasn't going far, I tended to wait for another bus (only a few minutes) if someone already occupied the wheelchair/buggy space as it wasn't worth folding the buggy for that.
I'm also well used to other babies on public transport, and while it's sometimes a little noisy, generally there are no issues with parents looking hassled and harassed off it. Often someone will try to interact with the baby or support the parent if there is a troublesome tot. It's life.