Skip the rack, panniers (? duh), lock, bags really of any kind. Get good cycling clothing. Jersey pockets are there for a reason, use them.

Spare tubes, Seat pack, Tire levers, Lights, Speedometer, Floor pump for home, Frame pump for the road, gloves

On computers, you can get a cateye wired computer for VERY cheap online (~$20) so why not. You'll at least know how far you are riding and know general speed data. My bike came with a computer and it honestly helps so much with making gains in riding because you can look down during a ride and know exactly how far you have gone. Maybe even try to pick up a used computer for even cheaper

Don't know if this was already mentioned, but try and not cheap out too much on things. Sometimes the money you save up front gets later spent replacing that product quickly. Or double after you decide to buy a quality product. Just a tip as new riders usually always try and find the cheapest thing rather than quality. That's what I did and was glad after I bought a quality replacement.

This is solid advice.

I would get a helmet, at least one set of cycling shorts/jersey, a seat bag, extra tubes, frame pump, multi-tool, bottle cages/bottles, and a cycling computer. That's the basics.

Edit: Tire levers!

Just riding around on the road? You don't really need lights, helmet, locks but they're nice to have. Essentially, the paraphernalia to fix a flat, water bottle and cage. The saddle bag is convenient, almost essential, for the tube, patches, tools etc because being attached to the bike you'll seldom forget them. I'd call the cell phone essential also. Strava app for the cell phone. Goggles, safety glasses or cycling sunglasses. MP3 player. Clothing accessories (jersey, gloves, shorts or bibs) are nice but also very optional.

For utility riding, and longer day trip type rides, you do need a lock. Some kind of bag to carry things, how to carry the bag is very much a personal preference. Commuting is a whole nuther story.

A light is always nice to have. Even if you're riding during in the afternoon, you never know if you decided to be out longer and get caught riding at night. I never ride without a light since I always ride at night now, but the one day I did because of a dead battery, I almost got into a head on collision with a car turning because he didn't see me. Went straight to the store to buy batteries that night.

Beginner's Accessories
 

A bell is always nice. People respond to ding-ding better than "hey look out".

I'll probably take mine off in the summer. I might keep the spare mag light in the seat bag.

A light is nice, and there are those who are convinced that it's a safety enhancement in the daytime but realistically it's just about pointless during the long days of summer, unless you do intend ride at night.

I'd add a tail light as essential equipment though. If you do get caught at night, number one priority is drivers able to see you coming up from behind.

Well yea during the day I don't ride with it. It's pretty ugly and I don't like to have too many things on my bike. But I start riding at 5:30 and by 6 I already need to have lights on.