I used to think that it was necessary to use a torque wrench to tighten the nuts securing the tire/rim to make sure they are tight enough without stripping the bolts. But this video https://www.youtube.com/watch?v=iJAI-cy2fZc doesn't even mention a torque wrench. I'd like to change the tires on my car myself if it's safe to do so with only the equipment that comes with the car.
While a torque wrench is not absolutely required, it's a good idea. You need to ensure the wheel is evenly clamped onto the hub. This prevents the wheel from warping and causing other issues. Using the lug wrench which comes with the car will get you down the road.
As a rule of thumb, I torque steel wheels to 80 lb-ft (50 lb-ft first pass; 80 on second pass, with a double check), while alloy rims will get 100 lb-ft (70 lb-ft first pass; 100 on second pass, with a double check).
You can get a decent torque wrench for cheap from the likes of Harbor Freight here in the States (not knowing where you're from). I've seen their 1/2" drive ratchet/click type ones go for as cheap as $11 on sale. Well worth the minor investment.
A torque wrench is not explicitly necessary; the spare jack and tire iron that come in the trunk of every car are sufficient to change a tire in normal cases.
If this was the case, car manufacturers would have to start including torque wrenches with the car in the event of a flat tire.
What the other answers failed to realize actually is that the wrench that comes with a car actually IS a torque wrench!
The length of the wrench is carefully chosen so that most people with average strength naturally torque the lug nuts with the provided wrench approximately to the correct torque.
Yes, it is a very inaccurate torque wrench but still it is a torque wrench. The reason why this is enough is that there is huge safety margin in the torques of lug nuts. 30% too low torque? No problem! 50% too high torque? Not a problem either! In one of the cars I owned, I accidentally looked at the wrong scale on the torque wrench and torqued to 120 lb ft (remembered incorrectly the torque would be 120 Nm), while the correct torque was 110 Nm. The torque 120 lb ft was 163 Nm, 48% too high. No damage done.
Edit: and a factor in the correct torque is also that most people have some amount of muscle memory. They have the tendency to remember how hard they had to untighten the lug nuts, and thus remember approximately the correct force when tightening again.