Now that we have already signed a contract, a question rised for me. Is it recommended to get mortgage from a bank or a private loaner?
This is exactly the type of thing that should be done before signing a purchase contract. The typical purchase contract in the US has a short deadline for the purchaser to arrange for financing.
There is no absolute answer to this. Your local bank/credit union might have a great rate. It might not. A mortgage broker might be able to find a great rate, but maybe not.
Now with 20% down then VA, FHA loans won't make a lot of sense but in some cases they might.
You will have to see what lender has the loan that best fits your situation.
The lender should be using the required disclosure forms, which will allow you to compare rates and fees.