AusFinance

1. Settle on Budget. How much cash you afford a month to pay, how much money can you put as down payment are the first try budget questions. Others such as your utility bills and other costs factor but I'm assuming you have some type of budget now, so you need to figure out how much you can afford a month and how much you can put down. Small note: depending your state, and the type of property you buy, taxes are a thing end of year. I can't offer advice on this.
2. Whatever amount of money you can put down should be 20% of what you're planning to buy. If you only have 20k, aim for 100k, 40k aim for 200k, etc.
3. Now that you have the money you can put down, take the total amount and go to any mortgage calculator. Put that amount and down payment in. That will give you a rough idea of how much your monthly payments will be. The number that changes is the interest. If you have a rough idea of what interest you'll get (from your bank or mortgage provider), the calculator will give you a rough estimate of your monthly. If you can't afford that number at all, plan to put down less money and buy something smaller or wait.
4. Why am I talking about personal finance first? You need to set realistic expectations. Banks and lenders have limits on what they'll let you buy but many will still let you buy something beyond your means. There's no reason to get your heart set on a place before knowing if you can realistically afford it or worse, convince yourself you can when you can't
5. Go to your lender of choice. I prefer large banks because I'm dumb. Smaller lenders may be able to get you a better deal but there are parts of the mortgage process I am unfamiliar with and my thought process has generally been it's easier to argue with a large institutions who fear public backlash. Again, this could be wrong, you've got to make your own call on who to borrow from.
6. Get a pre-approval letter. This letter essentially says that X lender is willing to give you up to Y amount of money. It's not a promise of a mortgage but it's saying the place is likely to work with you. It shows potential buyers you're somewhat serious and your possible income range.
7. Zillow, Trulia, local newspaper listings, word of mouth. Go find a place. Some people will hire a broker for this, they charge 1-6% of the sale cost, this is usually paid by the seller of the home. Some sellers will not work with a broker, some sellers have their own broker and that broker may not want to split commissions so they'll "ignore" your offer sometimes, sometimes your broker doesn't get you and is just offering you random shit on their books. Looking yourself is a lot of work. Paying someone else is easier but comes with downsides.
8. Go out there and visit places. Look around for property damage, leaks, mold smell, plug your phone into outlets, test your cell reception, look at the neighborhood, does it look safe is there garbage everywhere, etc. There are potential issues if you buy you'll then have to pay extra money to fix. Leaking roof? Few grand. Mold? Few grand? High cringe area? Potential break-ins and property loss. 9.If you have kids, this complicates matters as you'll want to be near a good zoned school. Same of you're planning for kids over the next few years. How far is your job? Do you plan to stay at that job? Do you plan on job hopping? Is your family nearby? Are you close? How healthy are they? What happens if you need to support them? What happens if they need to support you? All this needs to factor in before buying.
9. You find a place you love, it's perfect, and the price is right! Make an offer. You and dozens of others will make an offer. Some buyers just want the amount and pre-approval. There's a lot that the sellers can do here but for you, understand that your offer could get passed over. You can play games of offering a higher amount but be careful to not go over budget and be aware the person might be trying to fleece you for more money. Nothing you can do about it other than pay or... Move on. There are other homes out there. The home isn't yours until you paid, moved in and made it a home. Right now is an idea. Move on.
10. Get a lawyer. They handle the contract of sale between you and the buyer and deal with a ton of other back end paperwork.
11. Your offer is accepted. Now you've got to go back to your bank with ton of paperwork. This will vary but multiple back statements, multiple pay stubs, multiple years W2s, etc. Did your mom or dad give you a ton of money recently to buy? A notarized letter from them explaining that. Strange mark on your credit history? Proof of explanation of why. The bank will put you through the ringer because they want to make sure that if they give you this money, that you can pay them back. They will analyze a ton of stuff. Folks tout using mortgage brokers and they can help in many of these steps but keep in mind that is an added cost. Whether up front or in the form of a fee that adds cost to your mortgage. Some lenders may have fees for this portion.
12. The bank/lender will do their own due diligence and tests on the property. You can rely on theirs or pay out of pocket for your own as well. Is it up to code, is the property over baked, etc. The lender can come back and say "we won't lend as this place is a condemned heap" or " we won't lend because what they're asking is way more than what it's worth." You can try another lender at that point or go back to step 8.
13. The bank is willing to lend and the buyer doesn't back out. Closing. A date is set. Your lawyer will tell you the day, how many checks to bring and what amount, and be prepared to cry as your hand cramps up and sign so many documents, you'll feel like a pop star.

I'm pretty sure I've left off some details and there's a lot of nuance in some steps that can change your experience. The state you live in, the type of place you're buying all play a factor. Co-ops are cheaper but come with their own complications. Same for condos and houses, each with their own perks and flaws. Some folks start and close in 39 days, some people spend months or years in the process. There are those who will go drastically under budget because they value their monthly income, and others who will sacrifice day to day needs to live in their dream homes. If you're buying with a partner, there's tons of factors but the biggest two are being on the same page for finances and the home itself.

Also be wary of advice you receive. Consider who benefits. A realtor only gets paid if you close, so they may try to convince you of things. Your lender, if small, may need your mortgage so might lend or offer you advice to help close that may be in their better interest. Random person on the Internet? Who knows, maybe they're helpful, maybe they're deluded from their experiences, or maybe they're an idiot.

I wish you the best of luck and try not to get discouraged if things take a little while.

Edit: crap, forgot mortgage types. TLDR for most folks a traditional 30 year fixed is safest. If this doesn't sound right for you, you'll need to do a lot more research on your own first before talking to a lender.