How do I find details on rates and closing costs to refinance my mortgage with a VA Loan?

You can Google home refi and take a look at the rates and costs. Not clear what your objective is. Same payout but lower rate? Get some equity, lower rate, and pay the same monthly? These all matter. In general VA sponsored loans are 2-3 points higher than conventional. That's because the lenders have more hoops to plod through. A quick stop to the local bank loan officer can get you into the ballpark. BTW, the no closing costs options seldom are at the best rates.

Your credit score is the #1 factor in determining your interest rate. The above commenter is correct in that the things to shop for are closing costs and other fees.

VA simply guarantees the loan. Your rate and terms are still negotiated between you and the lender.
Therefore, how do you determine? You ask a lender. And then go mortgage shopping.
The home value, local market, your credit worthiness, available equity, current prime interest rate, all play a factor. Each bank evaluates or interprets these factors differently. And if you don't NEED the VA guarantee, compare nin-VA programs as well. (The primary gain with VA loan is no out of pocket at closing - no down payment and any closing costs are rolled into mortgage. If you have enough equity or cash to be able to finance down payment/closing on your own, you have more options.)
Do your homework. The more homework you do, the better your grade. In this case, your "grade" will be your rate and terms.
Don't just look at conventional 30 year loans, either. My first home I was able to do on a 15. Got me .75% lower interest rate. I pay just over an extra $100/month ($113) and will have it paid off in half the time. If you can afford to pay more than the minimum, look at 20, 15, or even 10 year loans.
Whatever you do, though, PLEASE don't do an ARM.