Newbie here. I've been riding for about a month now and have no experience riding in the rain. I have a Trek 1000 and was wondering if the chain, gears, etc... need any special attention to avoid rust. Does the bike need to be wiped down or is it ok to let it air dry.

Probably depends on how wet it got. Either should be fine, but you want to oil the chain (and maybe other parts) as soon as it's reasonably dry.

I also was going to ask, how probable is getting water in the axle bearings?
Late 70s Schwinn bicycle. Maybe some electrical tape around would keep it from water better...

I have a Trek 1000, too, and read somewhere the other day that (pardon my lack of knowledge of the proper names) something about the gears is unsealed and therefore vulnerable to rain and stuff in a way that better groups are not.

You can see how much I heeded the advice... but maybe someone can fill in the blanks for us. Haven't had to ride in the rain yet, so I just haven't worried about it.

if it is raining hard, it is possible that water will sip in into your hubs, bottom bracket, headset, etc. This problem is specially true for lower-end group sets, but is not totally elliminated even on high-end groups.

You can always detect this by checking how smooth your wheel turns, etc...

If you detect abnormalities(higher resistance), you should lube this components. Bearings with no lubrication can heat-up and eat-away the casing, and vice versa.

Don't lube it using WD40 or other spray lube products, it will only dilute the grease(solid lubricant) and cause more damage.

Well generally in a rain, water sprays all over your bike from the wheels and that carrys a lot of dirt. So your drive train and bike will get really dirty.

I spray my bike over with my water bottle several times and wipe it off.

I also clean and relube the drive train.

You can get by by relubing your chain but that means the dirt is all still on your chain to work as an abrasive.