I am a new, when I ride my road bike, it seems like I am damaging the rim whenever I go over a small crack in the sidewalk, and also, I seem to "bottom out" when I am taking a sharp corner, is this normal? I do not have an air pressure gauge just yet, but it feels by touch that there is enough air, even if it isnt at the highest pressure, could this also be my tubes? my rims?

Check your air pressure first. It is not normal to "bottom out" your tire anywhere. If that is the case you will eventually get a pinch flat, the easiest flat to prevent. Its really hard to judge air pressure in road tires so buy a pump or get a friend who has a pump with a gauge. Road bike tires should be topped off every few days to once a week.

I find trying to gauge tire pressure by feel is completely inaccurate, resulting in riding in under-inflated tires. If you have no pressure gauge, you can get an idea of adequate inflation by observing the tire deflection when you sit on the bike. The tires should barely have any visible deflection when you sit on the bike, this will probably feel rock hard to the thumb.

How often are you adding air? Bike tires need supplemental air to top them off every couple of days. A bicyclist is going to own a pump and gauge, that's part of the basic equipment.

Bike tires, particularly road bike tires are not like car tires.

Car tires run relativly low pressure, have thick tread and sidewalls, a large air volume and can go for months without losing significant pressure.

Bike tires run much higher pressure, have thin walls and a very small air volume. They have to be topped up every few days and squeezing them is totally inadequate to tell what pressure you have. +1 to getting a decent pump with a pressure gauge.

What kind of bike (road, mtn, hybrid, etc)? You'd be surprised how underinflated "hard" tires can be. I find with my road tires, they feel rock-hard to the touch at 60-ish psi, but they need to be inflated to 100-110, and anywhere from 60 up doesn't feel any different to the touch. And yeah, I made that mistake on one ride, hit a crack and flatted BOTH tubes (underinflation + impact ==> "snake bite" puncture). No cell phone, only one spare tube, long walk home.

What you really need is a decent quality floor pump with a built-in gauge.

On my road bike, it really starts riding squirrelly with the air pressure just down 10 pounds. Pressure really makes a difference.

Your thumbs are mis-calibrated. Get a floor pump with a built-in gauge.