We have 4.5k in savings.
This is probably not enough. You need enough money for the down payment, some miscellaneous costs (couple thousand), and the bank may require that you have a year's worth of payments saved up in reserve. That last part will depend on the lender.
We make 40K a year.
Rule of thumb is that you should buy no more than 2.5-4x your income as a house. Another rule of thumb is that lenders wont lend higher than a 36% debt-to-income (DTI) for the mortgage. Assuming a 4% rate, 5% down, and that taxes are 1.5% of the value of the property per year, that means they would probably let you borrow about $195k assuming you have no other debt. That's the maximum they would allow - but that would result in $1,200 payments per month, or $14,400 per year. That's a lot. More reasonable would be to keep the housing expense below 25% of your income, or about $10k per year, or a loan of about $135k at 4%.
Credit scores are, okay, 710 and 708.
You will probably want to wait a bit. The best rates can be had at 720+ for some banks, 750+ for other banks.
Do I find a house before I find a lender?
You will want to have pre-qualification in hand when you go to start looking for a house, and will want a pre-approval before making an offer. In my area, you will not get an offer accepted if you don't have a pre-approval.
I hear about FHA loans. Huh?
You need to do some actual research. Easily googleable questions are annoying to answer.
Should we wait and save more?
Yes. You need to have down payment, miscellaneous costs of a couple thousand, and a good emergency fund.
What are closing costs?
They are costs you pay at closing, which are separate from the price of the house.
I see things like 30y, 15y, 40y time lines. What?
This is the term over which you pay the mortgage. I.e., borrow $100k, pay it back over 30 years.
If we get a loan, say 200k and we get a house that's 160k. Can that 40k left be used for repairs and updates? Or is that money never used/issued from the bank?
The bank will not lend you more than the value of the house. Traditional mortgages will be issued up to about 80% loan to value (LTV). Special mortgages, like FHA, can have a higher LTV, but typically require that you pay mortgage insurance for the lender, which is an additional hundred or two per month.
So I ask you, help me understand, if even only a little.
This is a process that's very involved, and overwhelming at the beginning. However, it's a process that many hundreds of thousands of people go through each year. As such, there is a lot of literature and information available. Your post sounds like you didn't even google the basics.
CFPB guide: https://www.consumerfinance.gov/owning-a-home/
Investopedia guide: http://www.investopedia.com/updates/first-time-home-buyer/
Home buying guide from Discover: https://www.discover.com/home-loans/buying-guide/