Is this, what I should buy?
Yes.
I'd rather have everything I need to start out and not start buying more stuff in 2 weeks.
Where are you? Sometimes, it is better to buy stuff locally, if that is possible. You will probably change your mind on sensors more than once during the project. Waiting 2 or 3 weeks for stuff to be shipped to you can be frustrating.
Can this Uno handle all the sensors and outputs that I want to do, or would I need something bigger for that?
That depends almost entirely on the LCD display that you get. With a serial model, the UNO should be adequate. With a multiple wire model, it probably won't be. If that is the case, a Mega is what you'd want.