With my second I bought a lot less, because I realised how much didn't get used the first time. I would say...
Buggy-something that fits in boot and is light. Nothing too expensive as personally I swapped to an umbrella stroller around 7 months as easier and lighter and my £500 travel system suddenly felt like a waste of money
Car seat that fits on buggy makes life a lot easier
I breastfeed but got Mam starter kit-different size bottles and steralise in microwave.
Formula
Chico Next to Me Crib. Had a moses basket first time but much prefer this, I think she sleeps better as can see me
Joie 2 in 1 swing. My daughter loves it and has all her naps in there at home. Can also take it off and use just as a bouncer to move around house
Sling - I went to a sling library and then bought the sling I liked online after using it for 2 weeks. Some areas also have cloth nappy libraries or look at thenappylady.co.uk
Muslins/Vests/Sleepsuits (I got a couple of bundles off Facebook Groups)
Cellular Blankets
Swaddle blanket
First Aid Kit - Calpol, syringe, snufflebabe, teething gel/powder, thermometer, coconut oil for
Nappies (we use Aldi) and wipes (personally would stay away from anything fragranced, again we use Aldi)
At 3 months I've just bought
Next size clothes bundle (shes small and still in 0-3)
High chair
Playmat and toys
Jumperoo
Starting to look at cots
Good luck with everything!