First of all, welcome to the forum.
Second, congrats on the new bike. \\

/ I especially like the price. :shock:
Third, BassCliff will be along shortly with his little welcome bit, so, in the meantime...
Any idea if the solenoid smoked because it had not been used for a while and was merely burning some of the dust that had settled on it? Or did it smoke because it got stuck and was still trying to start the bike after you got your thumb off the button? Did the starter try to turn the engine at all? Does the battery have a good charge on it?
If the starter turned the engine, it tells me that the starter is at least not frozen up, but it might be a bit tight from sitting. Use an old screwdriver to short across the two large wires on the starter solenoid. That will turn the starter motor without using the electrics inside the solenoid. If it seems to turn the engine reasonably well, try it again, but with the key ON, full choke, NO throttle. I doubt the engine will start (unless it was prepped very well before storage), but that is what it will take to get it running. You can also try it again using the starter button. If it smokes when you use the button, the solenoid might need to be changed, but you don't need to spend $50+ on a new Suzuki item. Go to Lowe's or Auto Zone, get one for a lawn tractor. Same thing, will probably use the same mounting holes, only about $15.
In BassCliff's welcome, there will be a few links that you can follow that concern what it will take to get your bike back to prime running condition. One of the rites of passage will include a good carburetor cleaning and refreshing. Unless you ride very regularly all year long, that is something that is going to happen every couple of years, anyway, so doing it after a couple years of storage is almost a
must.
Another item that you will want to address rather soon is valve adjustment. There is precious little clearance to start with, and the valves tighten, not loosen, with use, so tight valves are a distinct possibility. Tight valves will also make the bike very hard to start.
Hope you don't mind getting your hands dirty, but be aware that there is a LOT of good help on this board. 8-[
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