Just bought my first house (moved in on 12/31/04) using the VA Loan (HUD, Convential, etc. should be similar). My process went something like this:
1. Make sure your credit is clear. When my lender's started pulling credit reports, it showed a negative entry (wasn't me and I'm getting it removed, luckily it was a minor item anyway and didn't affect my loan, but you might not be so lucky).
2. Decide what you can afford (be conservative). This ties into item #3. Use one of the many mortgage calculators (www.bankrate.com).
3. Check interest rates (www.bankrate.com). Do this religiously. They go up and down on a daily basis. You want to lock in your mortgage when it's low.
4. Decide where you want to live (which city, which neighborhood, etc.). I live in a huge metropolitan area (Dallas/Ft. Worth), so the options are insane. Take into account things like how far is it to your job, how far would it be to another job if you lost your current one, are there any planes/trains/automobiles nearby (pet peeve of mine).
5. Start looking for houses (www.realtor.com). Find about 10 that look nice and are in your price range. Now go drive by them and see if you actually like the way they look (on the outside) and the neighborhood. You'll quickly be down to 3 or 4.
6. Find yourself a real-estate broker (a buyer's broker - one that works for you, has your interests in mind).
7. Start looking at the insides of the homes you liked.
8. Start talking to different lenders, get the best deal (interest rate, closing costs, etc.). This is where you need to make sure you know the interest rates and compare the different closing costs (they'll give you a Good Faith Estimate, remember it's an estimate.
9. At some point, your realtor will want you to sign a contract with her (basically making sure you'll only do business for her). Remember, this is a two way street. My realtors initial contract had me locked into her for all of North Texas, etc. I crossed it out and put a single county. She was getting a 3% commission from the seller's agent (this is how she makes her money). The contract she wanted me to sign included wordage that I would be obligated to pay her if the seller's agent didn't (again I crossed that out). I also asked for and got a portion of her commission (0.5% of the selling cost).
10. So, now you've found a house you like and you want to make an offer. Hopefully, you've done some research (your agent can help you here). Find out what the house is worth by the property tax records (a lot of them are online now). Find out what other houses are selling for in the neighborhood, etc. Now make an offer. They go high, you go low and at some point you'll come to an agreement (hopefully in your favor). Remember, you are in the position of power in most cases (they want out of their house worse than you want into it). So, negotiate tough. The house I bought was initially listed for $5,000 below market value. Shortly after I looked at it (had decided I wanted it, but didn't let them know), I didn't contact them. After a few days, there realtor came back and said they were dropping the price $5,000. After my offer and a little negotiation, I ended up getting the house for $15,000 below market value.
11. So, they accept your offer. Part of your offer (which is a contract) should include an option out clause (you but the option to back out of the contract, something like $100 for 7 - 10 days). This gives you time to have the house inspected to make sure you're not buying a money pit (guess what, you'll usually have to pay for this - a few hundred dollars). Then after the inspection, you counter offer trying to get them to repair or lower the cost for anything the inspectors find.
12. Now, the inspections done and the house isn't a moneypit. You've made any changes in the offer you needed to because of the inspection. Your lender has approved your mortgage (you'll sign a gazillion pages) and now you have a closing date. On the closing date, you go to the Title Company and sign another gazillion pages (funny, I remember the huge tub of pens they had their). Wow, you now own your house and are in debt for 30 years. Congratulations!!!
Some other points to remember. The closing costs a lender charges you can be variable (different of lender to lender which is why you contacted several. You won't get the lowest rate shown on www.bankrate.com. For instance, if it's 5.34%, they'll give you a rate of 5.5% and you get the points credited to your closing costs (reducing your closing costs). Or you can buy points to lower your closing costs (you need to do cost-benefit analysis of this). Keep in mind that whatever you've saved up as a downpayment, it's not all going towards reducing the amount of your loan. For instance, you have $20,000 saved up and you're buying a $120,000 house at 5.5% interest. Well, you're not going to end up with a $100,000 loan because some of your $20,000 goes to paying for the closing costs (several thousand dollars). You may have $3,000 - $7,000 in closing costs, etc. It just depends.
For me, the whole process to a little over 30 days (I started house hunting during Thanksgiving holiday). It was nerve racking to say the least and I hate debt. But, I love the house and when I may the mortgage every month, it's towards something I own instead of a black hole.
I know I've missed a lot of steps, it's an overwhelming process. I'm sure other's have done this many more times than I have, maybe even someone here that does it for a living.
Good luck!!