I had a fold up travel kiddy toilet seat that I carried around for DD (she was little and that age too). It came in it's own little drawstring bag and fit well into a nappy bag, or I would just put under the pram (I vaguely recall the brand being Tommy Tippee, but could be really wrong there!). Think I just bought from Big W or woolies.

This seat also made her feel comfortable going anywhere, and like you said, we'd just take her as soon as she asked, wherever we were.

Now I just sit her on the toilet seat, but she was too small at 2 for that to be an option!

As for pull ups, we didn't bother, but totally up to you of course. I had her in undies at home, and fairly quickly she didn't want to be wearing nappies out either, so we just crossed our fingers and went with the undies - with a full set of spare clothes in the car, LOL!

Good luck and well done to your little girl for taking the lead on the toilet training!