You should buy all this stuff at once. Get a minimum of two spares, and tire levers.
1. saddle bag+mini pump+multi tool+patch kit+spare tube. This one is obvious, you cant go riding without it.
2. bottle cage + bottle. (Always get a bottle, hydration is so important even if your ride is 30 minutes. I have a 20 minute commute and drink about 400 ml)
3. Chain lube, etc for cleaning and keeping the bike in shape after a ride.
4. Decent pump at home, with pressure gauge. (One of the most important things I bought)
5. Shorts, maybe with bibs.
clipless pedals + shoes then Jersey is last.