As someone who's been through this 3 times, this all correct but I'll add a few caveats:

• If possible, go with a credit union over a bank. The interest rates may be a little higher, but you'll have fewer fees to deal with and generally have better service since CUs tend to be local while banks are national entities and couldn't give less of a shit about you. You'll have to be a member of a CU, but you can just open a savings account with as little as $5. Some CUs will also pay special extraordinary dividends each year to account holders.
• Get pre-approved for a home loan through your bank or CU and get the pre-approval for a little more than your budget. The pre-approval can help with negotiations and saves time on paperwork when your offer is actually accepted.
• If you pay less than a 20% down payment your mortgage processor will require private mortgage insurance (PMI) which will be added to your escrow. Your monthly mortgage payment will be divided into principal and interest (PI) and taxes and insurance, so you won't have to worry about paying your property taxes or home owners insurance each year: your bank or CU will pay that for you out of your escrow account.
• You will need to pre-pay 1 year of your home owner's insurance premium, so be prepared for that.

Buying a home can be a daunting, overwhelming, and frustrating experience, especially if you're looking at a quick closing timeframe. Just remember: your real estate agent is there to help you every step of the way and it's in their best interest to have your best interest at heart. Rely on them and their guidance and it'll make the whole experience less stressful.