I had a pram both times (second hand both times) because I walked for miles with both babies. I lived in a city and public transport was a faff so I was quite happy taking them out in their pram.
I used wet wipes when out but I had a top and tail type bowl which I stored cotton wool pads in one side with sudocrem and used the other side for putting warm water in. I still have the bowl and my youngest is nearly 11. I used it in the bathroom for storing bits and bobs.
I used nappy sacks as I was in a flat.
But both babies came in the bath with me from the start. It was just a lot easier and I loved having that skin contact with them in the water. I had bought a baby bath for DD1 but used it once. It was just too much hassle trying to fill and empty it and then it took up far too much space. I bought one of those seats which have suction pads and once they got too wriggly to hold and could sit up properly I plonked them in that and I sat at the other end.
Both had bouncy chairs too which were a godsend when you are trying to cook and need them near you but not under your feet! They would happily sit and watch the washing machine go round while I pottered.