Driving a manual transmission is almost somewhat of an art form. You need to fine tune your ability to combine your clutch pedal and engine at the same time. Every car is different as well. I hate driving Hyundai stick shifts because they have pretty sensitive clutches, so it makes it very hard to drive smoothly. I stall my Tiburon here and there, but I've learned over the years that every car is different. You'll have to find the sweet spot with yours and also get better at clutching and throttle control. It takes time. It took me about a half-year to figure out how to drive one without stalling it all the time. Now I can literally get in any manual car with no issues. You'll get it eventually don't worry!
As far as damaging the engine. Yes you can. You're putting odd types of stresses that weren't engineered to handle that type of jerking motion. I wouldn't consider it horrible stress though, if you drive it like that for years you'll probably mess it up, but if you learn you should be fine. As far as the transmission goes, it's the same idea as your engine getting messed up.
The real concern, is your clutch.
How a clutch works - Youtube
Riding the clutch is a term used to describe not fully engaging the clutch when accelerating. A lot of people have a tendency to "Ride" it by only engaging half way or not enough. After you watch that video, you'll understand why this is bad. You're essentially allowing it to spin just enough to move the vehicle. However when the two surface are not planted firmly and rubbing occurs for extended periods of time, it will wear and burn the friction disc and flywheel. Thus cause shaking and a slipping clutch. You'll end up replacing them every year. I've had a few customers that just don't understand how to drive one and usually come back for new clutches every couple of years.
Also, don't worry about hill starts and stops. They are tricky in a manual. There are several techniques, such as holding the E-brake and riding the clutch just enough to get a smooth start. I would recommend giving it a good amount of gas and slowly engaging the clutch until you feel it fully grab. Again, this takes lots of practice.
as far as 1st to 2nd. It's always a little rough on my vehicles. The ratios of gearing are usually very different. I find it to be that way in a lot of cars. 6 speed transmissions seem to be a little more forgiving, but 5 and 4 speed transmissions are finicky. I hope i helped you a little bit!