In order of things I bought:

1. Floor pump
2. Helmet (rule #64 )
3. Water bottles
4. Lezyne v10 multi tool (started with the v5)
5. Tire lever, extra tube and pump (practiced at home)
6. Jersey (to Stowe #5 off the legs + it's aero, welcome to the club)
7. Padded shorts
8. Higher quality padded shorts
9. Patch kit (never needed it yet)

Wish list:
10. Lights