I neglected a couple of important things about helmets. When it comes to impacts, helmets should be considered single-use. If they've ever been whacked, get a new one. The usual recommendation with motorcycle helmets is that they should be replaced if they're dropped. Bike helmets are so light that they won't hit with all that much energy after a short drop, so I haven't worried about this, but I'd caution you to exercise your own judgment. Finally, ultraviolet light will slowly damage the helmet. Avoid storing it in direct sunlight, and replace it after a few years of regular use.
My next recommendation is for a rear rack. This is mounted over the rear tire, and allows carrying things. Some small things can be bungee-corded directly to the rack, but your life becomes easier still with baskets or panniers. Metal baskets are inexpensive, fold up out of the way when not in use, and are the perfect size for grocery bags. You can attach them permanently with hose clamps, and, because baskets are cheap, they're not likely to be a target for theft. If you ever bike in the rain, you may prefer to get waterproof panniers. I like these, which are expensive, but they're big, have a roll-up top like a dry bag, and will probably outlive me.
I don't recommend just using a backpack, and recommend even moreso against a messenger bag. Your body has enough work to do -- let the bike bear the weight of your stuff.
If you want to leave your bike in public, you'll need to lock it well. Here in Berkeley, a lot of bikes get stolen due to the assumption that leaving it in a back yard or on a porch just for a little while was safe. It's not.
As with any theft prevention measure, you can't make stealing impossible. What you're after is tipping the balance of a potential thief's cost-benefit analysis from worth it to not worth it. This isn't that hard to do (and it's made much easier by so many people locking their bikes so badly.)
A bike locked with a cable lock scarcely counts as locked at all. A thief can get through one with a pair of bolt-cutters. Cost-benefit: worth it. The only locks I respect are good U-locks, and case-hardened steel chains with good padlocks.
Your U-lock should be case-hardened, and it should be one of the narrow ones. U-locks are vulnerable to attack from car jacks and lever arms. The narrow ones leave less room to fit a tool in; I don't recommend the wide ones. You needn't pay attention to bike-replacement theft protection guarantees. The fine print is that you have to prove the lock failed, i.e., present them with a broken lock. Thieves don't leave broken locks around as evidence.
Case-hardened steel chains are more versatile, and make it easier to lock a bike to a lot of things, but are even heavier. Kryptonite makes one. Personally, I use a quadrachain, which has a lock integrated into the chain instead of a separate padlock. And this is an intriguing article on a relatively cheap do-it-yourself case-hardened steel chain lock.
Coming up next: lights, visibility, and rain gear