Welcome, from another Washington resident (even though I live on the other end of the state)!
Did you read the link Mumma put up? Its got some good info about cycling. Yes, you can put the filter form the 5g into the 10g. It depends on what the filter is whether or not it will be enough filtration for the 10g.
Purchasing a test kit to keep an eye on water parameters really is the easiest way to know when the tank is ready IMO
. If you go that routre, make sure to get a liquid kit, not test strips because the strips are not accurate. I recommend the API master kit for freshwater.
Just so you know, most of the common fish you see are a big big for a 10g tank, but there are still lots of really cool options for a 10g tank that will thrive.