the 10 gal tank will be fine for a while.
As for a filter, you should get one that will be good for a much larger tank. For example, if you buy a filter that is good for a 10 gallon tank, you'll spend $25-40. Then later when you get a larger tank, you'll have to get a bigger one...thus spending money twice.
Just get a good filter...one that will be ok to filter about 100 gallons per hour. Many can be "tuned" down a bit for your smaller tank.
A canister filter is nice and quiet. If you want cheap, the types like an AquaClear that hang over the back have the best flow rate for the money, but will be noisy due to the waterfall they create.