I've tried avoiding riding during the rain for years because I didn't want to hurt my bike. But, tomorrow I really want to ride, rain (most likely) or shine. What should I do after the ride to ensure my bike remains in good shape?
Your title answers your question.
After the ride, wipe down the bike so that there is no excess moisture. Make sure that the bike is stored indoors, in a dry, warm environment, so that any remaining moisture can evaporate.
If you have a steel frame, consider using a frame preparation like Frame Saver, if you're going to ride in the rain regularly.
And last, make sure you do you regular maintenance. Open up all the bearing areas (hubs, headset, and bottom bracket) and clean and grease them regularly. Grease is protection from water and corrosion as much as it is a lubricant, so this is important to keep it running smoothly, as well as protect from weather.
I hope that helps.
What do you do with your car when it gets wet in the rain? The only thing you really need to worry about is the seat, and that's primarily because riding on a wet seat is unpleasant. (A shower cap is a handy thing to have to keep the seat dry.)
Re the bearings, the only bearing that's apt to get wet is the bottom bracket, which is a "sealed" cartridge on most mid-grade modern bikes. Opening up that area and greasing the outside of the cartridge and the inside of the housing will help prevent rust there. Otherwise, repacking bearings every 10K miles or so is sufficient.
You do need to clean the chain a little more often if you ride in wet weather, but that's about it.
I often ride in rain, so here is what I have observed and what I have been advised concerning post-wet-ride-bike-maintenance.
Advice: I have been advised to wipe the frame clean of mud, as mud promotes paint degradation and corrosion. Result: Most modern bikes use aluminium frames, those don't corrode (they oxidate, but ... nevermind). I haven't seen any practical benefit from this.
Advice: Remove the saddle post, turn the bike upside down and leave it like that overnight. Result: This is a MUST. Most times some water leaks out. This protects your bottom bracket.
If there is any mud or sand on the chain, front or rear sprocket, I wash all of them with a jet of water and a mop, then lubricate them. Otherways, there is a grinding sound and most surely, rapid wear of the parts.
Any further maintenance I leave for an yearly (what else to do in the winter) maintenance, consisting of complete disassembly of the bike and thorough cleaning.
Please, take this answer with a grain of salt. For example, despite following (2), the bottom bracket of my bike had turned to rotten rust for just three years.
Have fun! Riding in summer rain or in hail is and adventure!
For an occasional rain ride, the key (as others have noted) is to clean and dry the bike afterwards. Pay some attention to the drive train as the grit that accompanies riding in the rain can cause a sandpaper action quickly ruining your chain and causing damage throughout.
If you ride in the rain often:
One place where I differ from others is I do NOT recommend any of the "waxy" lubricants. These are marketed for the wet and cruddy conditions but it's exactly these conditions that make the waxy lubes evil. They have a propensity to hold on to grit and cleaning grit off a drive train that is well lubed with them is an undertaking. I'd recommend using a very light oil and get in the habit of a quick application after every rain ride and/or every hundred miles or so (go more by look rather than mileage). The lighter oils will actually help clean the drive train as you relube, wipe off excess oil along with some dirt and you are ready to ride again.
Get a good cycle stand, hoist it up, and clean it every week. Use the Park CG-2 chain cleaner and just give the cogs a good scrub with degreaser. Then wash off, let dry, and lube. Lube all the chain wheel as well plus all the moving parts. You will probably get slight rusting (very quickly) where there are screws facing upwards, such as on the handlebars. The rain has nowhere to go. So put a tiny bit of grease on the screw. Whatever you do, don't use a high pressure spray; it can clear out essential grease.