I am new to Taiwan living in Hsinchu. Please suggest where I can buy two-wheeler for commuting and what I must consider for buysing.
I am looking for something cheap or if anyone has it as giveaway.
I assume you mean pushbike/bicycle.
the www.formosanfattire.com site lists bike shops, dont know if there is hsinchu info tho… sorry.
decent bikes can be head pretty cheaply… no real tricks when buying cheap other than stick with a reasonably decent brand like giant or merida.
I mean motorbike like scooty!
you need help finding places to buy scooters??!?! seriously? check the 4 sale section here - they sometimes come up.
I’ve said it before and I’ll say it again…rent…I do and I love it…I have one of those fake Harley thingies…I call it Piglet…I pay 2500NTD a month and I don’t have to worry about maintenance or theft or even a driver’s license…oooops…not that that’s a problem MR. Policeman…but if you had the guts to try and speak English to me, you’d find out for sure… 
(knocking on some wood)
PM me if you want the guy’s number.
There are a bunch outside my building. Do you know how to hot-wire?