Would riding in rain, through water puddles or wet grass, or mud do anything harmful to my bike? I have a cromo frame, and cromo cranks, and everthing else is some kind of steel.

If the steel is treated propperly then the answer is no. Mud is certainly the most likely to cause problems, due to its ability to hold water, but internal frame rust can be a greater concern in the long run.
Externally, I prefer a good carnauba wax. It looks great and it works really well at sealing the paint and any scratches you may have.
Internally, Use a rust treater like Extend or the like(there are several products on the same vein) and follow it up with a coat of LPS-3 spray-in sticky oil. There are several threads on the site related to the matter, search 'internal frame rust' 'frame drain hole' and a few other threads may pop up.

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Well I have a BMX bike, I don't mountain bike. So I'm not like constantly going through mud or anything like that, I mostly just get some wet stuff on my bike that dries. But when I come to clean my bike off, if I get a little damp cloth it all comes off real easily. So am I fine, or do I need to still watch out for internal rust?

PS. I usually try not to ride when it's really wet, and when I do I avoid going through wet spots and such if I can.

yea it will damage your bike but to keep from that after you get done riding rinse the bike off to get the mud off and then get a dry rag and dry it and it won't rust or nonthing!!!

Hows about rust on the inside. Like army food....kills from within.

if you go through puddles like FlyBoy said surely water ain't gonna get in the frame that easy!

Flip the bike over and drain the tubes if really wet.

Use lots of oil, on the seat post, chain, yer cog. You might find a few cheap bolts rust, replace with better ones.

I hate rust in my hex bolts.

Ok, thanks alot guys, I usually wipe my bike off anyway after I get it wet, so I should be fine. Thanks again!

The place the water mainly gets in is the seatpost clamp, so if you get skunk striped in a puddle, more than likely you have some moisture. Whether or not its a lot is another matter. The simplest approach if you do it carefully and don't want to go the full route previously descrbed is to take a WD-40 with the red straw and insert it into the frame drain holes. A little puff is all that is needed. DO NOT do that down the seat tube however, since that can get your bottom bracket(assuming you have open type bearings).

Ok, but why wouldn't I want to get it on my bottom brakcket? And also, does WD-40 do anything bad to your bike at all? I mean anything at all? Because it really works good to clean off the frame when the stuff is crusted on, so I usually spray some into a cloth and wipe off the frame. But I didn't know if it did anything bad to any of the parts on my bike, or the paint. Thanks!

Mud will make your brake pads wear pretty fast. So keep an eye on them a little more than usual. Shimano pads are really bad for this. Kool Stop last much longer in the rain.

WD-40 is a solvent mixed with a light oil. Thus it acts on rusts and lubricants to soften or eliminate them. It will dissolve bearing grease if exposed such manner as spraying down a seat tube could cause. The major thing to watch for with any WD operation near a bike is overspray. It oversprays a mist that can get on rim braking surfaces and brake pads, rendering them effectively useless. Treat that with caution. Paint shouldnt care.

Ok, thanks alot! So paint don't care, and watch out for bearings and lubricants and stuff like that, but is everything else ok then? I'm not saying I bathe my bike in WD, but it is a really good cleaner and a pretty good lube for SOME things. But anyway thanks for the headsup on the bottom bracket thing, but or else like you said everything else should be fine, right? Thanks!