Why is the fluid on my dipstick clear after an oil change?

That clear fluid you are seeing is almost certainly the new oil before it has had time to mix with the used fluid that was sitting in the oil passages, so it is not the transmission fluid. The starting issue is the more immediate concern, and usally indicates a problem with the fuel system, ignition, or compression rather than the oil itself.

If you drained the wrong plug, you would likely be facing an immediate catastrophic failure, but since the car cranks strongly, the block is prolly fine. For a 2004 Corolla, the oil pan plug is located on the very bottom of the engine block, so it is unlikely you tapped the wrong thing unless you have severe body lifting or aftermarket modifications. Assuming you are working on a standard setup, you just did the change correctly in terms of the plug location. Since the car cranks but will not fire up, don't waste time topping off the oil and trying to start it again; that won't fix the lack of ignition. First, you need to verify you have the basic engine requirements met. Start by checking your battery terminals for any corrosion or loose connections. Sometimes, a poor connection can make it seem like a starter issue when it is actually just a power supply problem for the fuel pump or ignition coils. Next up, you need to check the fuse box. Find the fuse related to the fuel pump and swap it with a known good fuse. If the pump is dead, it won't start regardless of the oil condition. If the fuses are fine and you confirm power is going to the engine, then you'll need to check for spark at the coils. The combination of cranking but not firing means either no gas or no fire, but checking the easy stuff first saves you from taking the engine apart.

One more thing to try before you chase spark and fuel, pull the oil cap off and look inside with a flashlight. Back in the day you could tell a lot by that. If it's dry in there, you're fine on oil. If it's full and brimming, you double-filled it somehow. Also on those old Corollas, check that you actually put the drain plug back in the right hole and snugged it up. I once watched a neighbor lose a whole pan of oil out his drain plug two miles from the house because he'd cross-threaded it. Smelled burning oil for days. And a dumb one that's bitten me twice, make sure you didn't knock a vacuum hose loose leaning over the engine. Those rubber lines crack with age and pop right off when you bump them, and the car won't run right or at all.

Update: I pulled the oil cap and shined a light in there like one of you suggested, and it's not brimming full, so I don't think I double-filled it. Dipstick still reads okay too. But the car still just cranks and won't catch. So now I guess I'm chasing the spark or fuel side of things. What's the easiest way to check wich one it is on this car?