Here's the thing: you can't make serious offers without a pre-approval letter. Sellers won't take you seriously, and agents won't invest time showing you homes if you haven't proven you can actually get financing.
Pre-approval is different from pre-qualification. Pre-qualification is "we think you might be able to borrow this much." Pre-approval is "we've reviewed your financials and we're willing to lend you this amount."
Find a lender
Talk to a mortgage lender. We can recommend good ones if you don't already have one. Who you hire matters.
They'll pull your credit, review your income and employment, look at your debt-to-income ratio, and verify your assets. Then they'll give you a letter stating how much they're willing to lend you. This takes a few days, not weeks.
What you'll need
Recent pay stubs
Tax returns (usually two years)
Bank statements
Employment verification
Credit history
If you're self-employed, expect to provide more documentation.
Downpayments
The lender will strategize with you on the best loan types for your financial situation and approved interest rate. Downpayments can be as little as 3-5%. You may be able to reduce your monthly expenses if you can put 20% down. A good lender will help you make sense of it and understand exactly how much money you will need.
Protect Your Pre-Approval
Once you're pre-approved, don't do anything that could jeopardize it:
Don't open new credit cards
Don't finance a car
Don't make large purchases - like furniture and appliances (wait until AFTER you close)
Don't change jobs if you can avoid it
Lenders re-verify everything before closing, and any major changes can blow up your approval.

