Starter spins freely, will not engage

Wow. Probably the positive and negative connections have gotten reversed in the solinoid and starter motor. Still stuck? I recommend a DIYer's repair manual that shows electrical connections for the starting system.

I need to know what type of starter motor yours is

The starter engages with the starter clutch, normaly when the starter spins and doesnt engage, it is because the clutch bearings within the clutch have worn the machined surfaces causing the clutch to stick and not engage when the starter is pressed

The starter clutch needs to be replaced. At the other end of that chain is a wheel with a couple slots cut at angles. Inside each of these slots is a small cylindrical piece of metal. The whole unit fits over the end of the crankshaft and because of the angle of the slots those cylindrical pieces of metal (which move freely in the slots) will be forced against and grip the crankshaft when the starter is operated. If they become worn down they are no longer able to exert enough pressure on the crankshaft in order to turn it. Buying a starter clutch 2nd hand is not recommended. If you are unable to source one from a bike shop, try and have them made. Take the old ones to an engineering shop. Unless you know how to temper/harden metal of course.

Check fuses. Check voltage. Could be a burned, or just broken fuse. Could be a dead battery. Could be as simple as poor battery connections. If not, perhaps the following will help:

A starter can be tested in a very simple way, by applying 12vdc to the power input, and grounding the case. If it spins, then it's on to step 2, checking connections.

Starters are much too simple to be more complicated than that, but the electrical system can be. Check the circuit between the relay and the starter. Is is a direct wire with no breaks? If so, check the voltage at the power output and input of the relay. If it is good, check the power input and grounding of the starter.

Most new bikes have a clutch switch, and it may interrupt the 12vdc to the starter, without affecting the relay. Same thing goes for the nuetral switch.

Try starting in nuetral, with the clutch pulled. Check the relay for conections that you may have taken off, and not replaced. Same thing for the starter.

I would guess it is the starer clutch, also called the one way clutch. I went on to www.bikebandit.com and looked up your bike and it has a nice exploded veiw of your starter clutch . I will try to put a direct link to it , hopefully it will take you to it. http://www.bikebandit.com/houseofmotorcycles/yamaha-motorcycle-tt250rl-1999/o/m7707
if not go put in your bike and look under starter clutch.

The intermediate cog will spin freely one way but not the other, works on a principle similar to a centrafugal clutch.