Radiator problem in car but not overheating?

I have a 2001 Honda Civic with 130,000 miles on it. Earlier this week I noticed it was smoking white smoke from where the radiator sits in the car. I put some coolant in it and I've noticed that the reservoir of coolant gets low within 1 day of me putting coolant in it. My heat works and my temperature gauge is normal so I know it's not overheating. Someone suggested a block in the radiator, something about getting it flushed out. I was told that expensive. I'm in college working part-time so I don't really have that much money.

If it's not overheating, it doesn't need to be "flushed out". You probably have a small leak somewhere, like maybe one of the radiator hoses. Get it looked at.

You have a leak in your radiator

Mentioning your financial situation is pretty much irrelevant. Your purpose for being here is to get an answer, hopefully, and not to inspire pity. Car repairs can be expensive, especially if you don't perform them yourself. This is something you should be saving for every single week, as you drive a 15 year old car. Driving a 15 year old car should come with a basic understanding that in our universe, things break, and it happens more often with age. So take $20 out of your weekly beer fund or whatever and put it towards car repairs that WILL come up.
That having been said, coolant can only go so many places. You need to find out where NOW before those problems cascade into OTHER problems. For instance, if you have coolant leaking out of your water pump shaft, that'll cost money to fix the water pump. If you let it go and the bearings in the pump seize, you just bought yourself more repairs you can't afford. So coolant usually leaks: From the radiator; from a hose somewhere in the engine bay; from the heater core; from the water pump; through a hole in the head gasket that has blown and is flowing either into the engine or out of it.

The first thing to do is the most obvious. Look for leaks. Open the hood, if there are trim panels covering the radiator or top of the engine, remove them. Remove any cover under the engine. Get a good flashlight and look for leaks. Check for a puddle on the ground after the car sits a while. Start the car up and let it get good and hot and see if you see any venting in the form of steam or a jet of liquid coming from the radiator or a hose. Some hoses go clear into the firewall (for the heater core) so follow and check them all.

If you don't see anything obviously leaking, then chances are it's leaking into the engine through a head gasket leak. It'll either be going into a cylinder (and being combusted and turned to steam in the exhaust, which you can sometimes smell if you give your exhaust a whiff) or it'll be leaking into the engine itself. Now would be a good time to drain your oil and change it. Check the oil and see if it has any liquid or milkiness to it. Also check inside the radiator and see if any oil is present there. If you see no oil in the coolant or vice versa, chances are it's going into a combustion cylinder. Before you go yanking off the head, do a compression test. One or two of the cylinders should have low compression or difficulty holding it. If you see an obvious compression issue, then you have your culprit. If you find it is a head gasket, do this repair CORRECTLY. If you cheap it out, you WILL do it again. Take the head to a shop that does engine machine work and have them check it for flat. A cylinder head must be within x number of thousandths of flat. If it's warped ever so slightly beyond spec, a new gasket will simply blow again. If it isn't flat have it machined down, and shimmed if necessary. Replace the gasket with a high quality replacement. Use NEW head bolts. Don't reuse the old ones to save $12 or, once again, you will be doing this repair AGAIN. Torque them properly and in sequence. Specs are available online for your engine.

Whatever the problem is, remove all the old coolant, flush the system, and replace it with fresh. If you mix your own coolant vs buying the 50/50, make sure you use distilled water not tap water. Or you will be doing this again. (notice the repeating theme of doing it right might cost you more now but less later?)

Definitely sounds like a leak, and probably a pinhole in a hose that only opens up when the hoses get hot and flexible. I say this because you mention smoke. What you're seeing doesn't sound like smoke, but steam from the coolant spraying onto the engine. Coolant leaks are usually hard to spot, the easiest way is to catch it in the act, so next time it smokes, pull over immediately and with the engine still running, open the hood and see if you can tell where the leak is.