Car steering wheel shakes a bit, can feel it through floor sometimes?

The car being out of alignment is wearing the tires out and they are out of balance.

Go to the Hunter GSP-9700 website. As you click through it there will be a box to enter your zip code. After you enter your zip code a list of private mechanics, new car dealers and tire shops who have the Hunter GSP-9700. This wheel and tire balancer is by far the best in the world today. It may cost no more or slightly more to have the wheels and tires balanced on this machine.

M has given best answer. there are several reasons that this can occur. pulling to right is often due to damaged steering components or out of align chassis ( accident /kerb damage) it can also be due to badly worn ball joint etc

vibration is most common for out of balance tires. other factors can induce this too. failed damper(s) play in ball joints .badly worn tires . excess wear in steering mechanism including steering bush etc

only a proper fix will cure things. have a garage check it over. driving a car pulling to side is a sign of unroadworthiness and in any accident a claim may be refused! its worth spending a few bucks to get car right and keep ourself ( and others ) safe on road.

An alignment and tracking check will help with the pulling to one side, but if you notice a shimmy through the steering wheel I'd say your rims are out of true. Heavy impacts can bend aluminium wheel rims out of round.

Vibration is about an out-of-balance tire. Possibly a defective tire.

A misaligned front end (steering) pulls or wanders and/or eats tires.

You need both. Probably due to wear of front suspension and steering parts.

You need to get a wheel alignment and balance done. Do it sooner rather than later. Having the alignment out can ruin new tires in a few days.

Ordinarily, alignment won't cause a car to shake. But there are a few exceptions and your isn't one of them. You have a crooked belt in one of the front tires. Swap the tires around to see if it makes a difference.

Have the front wheels balanced at the same time the alignment is done. That should smooth out the problem. Your tires will last longer too.

Get all four tires balanced plus an alignment.

You need an alignment, and you need to have your brakes checked. No excuses - other than an oil change, there aren't many maintenance jobs that are cheaper than an alignment.