What are some big mistakes that first-time home buyers can make?

1. Overspending.
2. NOT using FHA loans, military, etc.
3. Not having 6-months of TOTAL estimated bills in a do not touch unless you absolutely must savings account. Put this in a bank all by itself so your bank can't auto-withdrawl if your checking account goes empty.
4. Not saving for maintenance. Even if it's a new build. You will have maintenance costs. Hopefully just lawn care.
5. Don't get a home with large square foot (or meter) mulch beds. God I hate mulch. This is my one almost regret. If I were to put down new mulch, 1" thick (should be 3"-5" thick), it costs me $600 per year. Not including labor.
6. Not making sure your roof, windows, doors, HVAC, electrical are better than good. Your typical inspection will just give you a yea/nay. You will need to pay for more detailed inspection, especially if you want to know an estimate of how long they'll last before you need to replace them.
7. Not checking flood plain maps and asking existing neighbors if their basements flood, how often and how much.
8. Not verifying your property line, fence line, utility easements, etc. Don't trust your neighbors or the GIS website mapping (that's a rough estimate and NOT certified or usable in court).

Might I ask what country you're planning to buy in? Different countries would have different regulatory environments and different financial schemes, which would influence the advice people should give you.

You have presumably rented before, so must be aware of the basics.

Most home owners get distracted by shiny appliances, staged furniture and a fresh coat of paint.

Spend some time looking at the roof (is it new), the foundations (any cracks), the circuit breakers (is it upgraded) and the plumbing (I like copper).

Also, I am a huge fan of crawl spaces and attics.

I'll add:

1. Don't believe the bank when they tell you what you can afford. I found a huge difference going to a bank and going to a credit union. When I got my first mortgage about 10 years ago, a bank told me I could carry a mortgage of $550k or something ridiculous like that (I actually laughed when he told me that because I assumed he was joking). The guy at the credit union, in contrast, "strongly cautioned" me about anything above $250k. When a bank tells you that you can carry a big mortgage, they are thinking about themselves: they are not thinking about you. If you follow their advice, you are going to be completely fucked and they are going to be laughing all the way to themselves.
2. Don't assume interest rates will stay the same. Give yourself a bit of a cushion in case they go up.
3. Don't go for the fixed-rate mortgage. Does this contradict #2? No, if you follow #2, you don't need a fixed-rate mortgage. My dad carried a mortgage through the crazy 1980s (when interest rates suddenly spiked to 18%) and he said that even through that crazy period, he would have ended up paying more if he'd locked in to a fixed rate when rates were low. A variable rate will always mean you pay less overall.

1. Get the home inspected by a licensed home inspector.

2. Don't purchase more home than you need.

If you buy an older house 60s-70s make sure to check if it has aluminum wiring and if so to make sure that all the outlets have been pig-tailed to copper. Aluminum wiring was popular in residential construction during that era and the exposed ends tend to oxidize and catch fire. Copper rust still conducts electricity well enough, aluminum wiring does not which cause the wiring to heat, which also causes it to expand and then when it cools it can come loose which is a recipe for fire.

If it's been properly pig-tailed to copper then the risk is pretty minimal.

When you sign the final papers at closing, they don't tell you about all the spam mail you'll get from companies posing as your lender trying to sell you mortgage insurance. They grab your info from the newly filed public records then disguise mail to look like it's coming from your lender. I thought this was something I needed to fill out and return to my lender because it had their name/info at the top. Next thing I know, I have solicitors knocking at my door trying to sell me something. I brought this up to my lender after the fact who said "someone should have told you"... yea, no one told me. I had to block a bunch of phone numbers when solicitors kept calling even after being told no. Really infuriating.

I'm from the UK but this advice should still be valid.

1. Get a full survey (includes structural elements) done by somebody qualified & independent.
2. Just because the bank will approve a mortgage up to a certain amount does not mean that's a good idea or goal to aim for. You should probably go quite a bit lower.
3. Check flood maps & see if neighbors have had problems before.
4. Test water pressure & water quality.
5. Check the house + any additions are allowed to be there (in the UK we'd call it planning permission).
6. Likewise boundaries/deeds need looking at closely, plus access to the house (shared driveways come to mind).
7. Take what estate agents say with a grain of salt.

Since you're in the U.S., get a subscription to Angie's List for the area you're planning to buy in (it's worth it to find reliable contractors for all kinds of work).

Hire an independent building inspector to review the house before your purchase.

Not only will this disclose issues for a house that's only up to code on paper, but you can use the information in price negotiations, and plan some of the future maintenance investments.

Whatever you do, make sure you get a proper survey done. Learnt that the hard way!