Once you have a rough sense of what will meet your mission (e. g. new 172, pre-restart 182, 152) you need to figure out how much it will cost you; not just to acquire, but to operate.

Fortunately, getting a good sense of what you’ll pay is pretty easy. You’ll have some initial fixed costs (in addition to the cost to acquire):

One-Time Expense How to Estimate
Escrow Fees and Title Insurance Call a title company that deals in aircraft sales, and ask for a quote. The price will depend on the plane's cost.
Aircraft Cover Bruce's is the gold standard. Their website has a price list. They usually give out modest discounts around Oshkosh and other large aviation events.
Sales Tax Ask an account familiar with aviation issues, or call your county tax commissioner. You may be able to get a casual sale exemption.
Misc plane accessories Depending on what the previous owner throws in, you might be in the market for GATS jars, chocks, windshield cleaning equipment, and so on. Sporty's is the place to go for such sundries.

Then there are fixed costs you’ll incur every year, regardless of how much or how little you fly.

Yearly Expense How to estimate
Tiedown or Hangar Call your airport, or better yet call a few airports to compare
Insurance Call AOPA's insurance hotline. If you tell them a bit about yourself and the plane you want to get, they can give you a ballpark right away.
IFR GPS Data Check Jeppesen's website.
Annual/IFR inspections Call a few local mechanics. They'll give you an estimate if you tell them the type of plane.
Ad Valorem Tax This will be a few % of a fraction of your plane's value. Ideally your local county will have a rate table online; failing that you can call them on the phone.

As much as I’d like to, I can’t provide a ballpark estimate for your case because all of these vary so widely by pilot, aircraft, and location. However, here’s a representative example for a pre-restart fixed-gear 182 based at a towered airport flown by a single IFR-rated pilot:

Expense Amount
Tiedown $90/month
Insurance $1,400/yr
IFR GPS Data $500/yr
Annual/IFR Inspections $2,000/yr
Ad Valorem $1,500/yr
Total Yearly Cost $6,480/yr

Next, let’s figure out the hourly operating costs. These are the hardest to figure out, but we’ll do our best.

Hourly Expense Amount
Fuel Look at fuel burn rate in the POH, and multiply by the cost of fuel in the areas you plan to fly.
Oil This will be $2-4 an hour
Prop reserve Ask a mechanic how much it costs to overhaul a prop, and divide by the prop's TBO
Engine reserve Your plane's engine will need to be overhauled very rarely (probably less than once a decade). However, when it happens it costs more than $25,000. You don't want to be in a situation where you have to delay an overhaul for financial reasons; that's very dangerous. To avoid this situation, for every hour you fly transfer some money to a separate savings account so it's ready when you need it. To estimate the amount to reserve per hour, divide the cost to overhaul (probably around $25,000) by the engine's TBO (1500 or 2000) hours
Misc MX reserve Things other than the engine and prop will periodically break and need repairs. Call a few mechanics and ask for their input. Hourly fuel cost / 4 isn't a bad starting point.

Returning to our 182 example, you might get something like this:

Hourly Expense Amount
Fuel $80
Oil $2
Prop reserve $2
Engine reserve $19
Misc MX reserve $20
Total Hourly Cost $123

For bonus points, you can also consider the costs per trip such as landing fees or customs use fees. However, even at a busy towered airport, they will generally be less than $100 so unless you fly almost every day it won’t move the needle.

Finally, combine the fixed operating costs with the variable costs for a range of realistic flight times per year. This will give you total operating costs. For our example, that works out to:

Hours per year Total operating expense Cost to operate/hr
50 $12,630 $252.60
100 $18,780 $187.80
200 $31,080 $155.40

A 182 seems to rent for >$200/hr wet these days, so you can see that after several years of flying you’ll break even, assuming you fly often. Also, remember that only the fuel is an immediate out of pocket expense; the maintenance reserves will be earning you interest and might not come due for many years if you buy a plane with a good and/or low-time engine.

At this point, you are ready for the scariest part of buying a plane: checking your finances to verify that you can afford a plane, and getting your family on board.