Knowing where the water lands gets you in the area, but what you really need to know is where the water came from. Is the tank cracked, is the supply line failing, is there a bad connection on either end of the supply line, or is the valve itself going.
If the tank is cracked, you'll have to buy a new one, and it's a large project if you're not used to working on basic plumbing jobs or are skirmish around an open toilet.
A bad supply line can be easily replaced and it's even easier if it's just the washers.
A bad valve may just have some dirt or buildup in it that will be fixed by opening and closing it once or twice with water running through it (an empty toilet tank). Though keep in mind any dirt may then get in the toilet valve and could cause it to run slowly until you clean it out. If that were to happen, just post another question. It could also be the packing material in the valve, which I've never touched on this type of valve. If that's the case, you may be able to minimize it by opening the valve all the way. If the valve is too far gone, you'll likely need a blow torch and solder to replace it. This can be a DIY job, but if you get in over your head, the water for the whole house will need to be shutoff until you can get a plumber out.
If the drip becomes a flood, it's not the cost of the water you should be concerned with; it's the water damage.
You say the leak seems to be coming from the valve itself, and your reasoning makes sense; if the toilet's dry it means the valve is turned off, so it's probably not any fitting beyond the valve stem. This means that the shutoff valve gasket, or the valve itself, has worn out or corroded and needs replacing. This happens over time; just like the flapper valves in the toilet and other elements of your plumbing, the shutoff valves corrode with acidic water, calcify in hard water, deteriorate in highly chlorinated water, and just plain wear out from use and water flow of any type over the years.
This is definitely something that nobody would think less of you for if you called in a plumber. However, it is DIY-able. You'll need to shut off the water to the whole house (there should be a shutoff valve near your water meter, wherever that is, and you may need a special tool to turn it off), then bleed the remaining water out of the interior lines (easy as turning on the faucet on the sink to cold, or flushing the toilet a couple times). With a bucket or pan under the valve to catch any remaining water in the line, disconnect the toilet from the valve and open the shutoff valve, letting any remaining water drain out of the line and valve. If this is a first-floor bathroom, there may be quite a bit to drain as the upstairs supply pipes will drain out as well. You can help it drain faster by opening a couple of faucets upstairs (providing a vent for the pipe stack similar to what you have in drain pipes). If it never seems to stop, you probably didn't completely shut off the outside valve.
Once the line is drained, you can work with the valve. The first thing to check is the fitting between the valve and toilet, since you have that apart anyway. Check the gaskets/washers on either side of the fitting. Some are compression fittings, and work by sealing a copper or nylon line against a ball inside the fitting; it won't work well if the end of the line or the ball has deep gouges, scratches, or is generally deteriorated. Most toilet lines are steel-braided rubber with threaded fittings, and usually can be easily replaced if necessary.
The next thing to check, and much more likely to be the problem, are the gasket washers in the shutoff itself. There may be one or two depending on the valve's design; one to seal off the supply line, and another to make the valve stem watertight. The valve stem should be going through a fitting screwed into the valve housing; unscrew that with a crescent wrench and you can take the whole thing out. You can usually also unscrew the handle from the stem and completely disassemble the valve stem. Clean off any calcium deposits from the washers, and inspect them for extreme wear or other deterioration. You can buy a bag of these washers for a couple bucks at your local hardware store. Reassemble the valve with the new washers, close the valve completely, and re-pressurize your house plumbing (make sure to turn off all the faucets you opened to drain the lines) to see if that did the trick.
If that didn't stop the leak, or the valve stem or body appears highly corroded and/or calcified, it may be time to replace the whole thing. With the house shutoff closed again and the lines drained, look at the valve's connection to the wall pipe. There should be either a compression fitting or a threaded NPT connection between the two, and in that case removing it simply takes your crescent wrench and some elbow grease. Some valves, however, are soldered directly on. If this is the case, I personally would stop and call a pro, but if you're handy with a soldering torch you can get the valve off yourself (careful you don't set the wall behind your toilet on fire). Then, simply take it to your local hardware store and buy a replacement with the same fittings; it should cost you about five to seven bucks. If the valve was threaded onto the wall pipe, get some Teflon tape as well. Use at least four wraps of the tape around the threads of the wall pipe (put it on in a clockwise direction, and stretch it a little to get it to mold to the threads), screw the valve firmly back on to the pipe, then re-pressurize the house again with the new valve closed (and all other faucets off) to check for leaks. If there are none, connect the toilet supply hose, open the shutoff, and make sure it still doesn't leak during and after the toilet fill-up.
If it STILL leaks, call a pro; you may have missed something, or done some damage in removing various parts.
Best Answer
The original leak is most likely either caused by a failure in the rubber washer that connect your supply line to your fill valve, or a leak in the seal that connects your fill valve to the tank.
This is a common problem, especially with older toilets that use natural rubber washers. The rubber breaks down over time and will eventually begin to leak. The newer synthetic washers will last much longer.
It is unfortunate that you already have taken it apart because you will have a harder time determining the source of the leak. When you run into a leak of this kind, it is best to first check the supply line itself, as it is the easiest to access. Shut off the water supply, and drain the tank of water. Then unscrew the supply line and inspect the washer. It should be firm and flexible, yet hold its shape and be in one piece. If it is not, then it needs to be be replaced and you can reinstall the supply line. After a quick test, you should be able to see if this fixed the leak. If it leaks from the supply, make sure that the washer is in place. On Occasion, the washer is missing right from the beginning. They either fall off, or people take them off.
The other place it could be leaking is from inside of the tank. You will need to disassemble the toilet as above. You will then needs to loosen the retaining nut on the outside bottom of the tank where the supply line goes. Loosening this nut should allow you to lift out the fill valve. Inspect the washer on the valve, and check if it is good. Also check the threaded part to make sure there aren't any cracks in the fill valve body as well. Replace washer, and reinstall.
By replacing both of these washers, it should solve the leak. If it doesn't then there may be a fault in the parts themselves - either the supply kine is bad, or the fill valve, or possibly a combination of both. Also be sure that it is leaking from the supply, and not from anywhere else. There could either be a crack in the porcelain, or there could be a bad tank to bowl seal. There are "universal" rebuild kits available out there that include everything that you need to rebuild a toilet. They include all washers, fill valve, and a flapper. The only part that is not included is the supply line. If the internals of the toilet are old, or not functioning properly, then it would be a good idea to do an entire rebuild.