Can a Arduino Uno handle all these components? Its my first project with an Arduino, so please bear with the dumb questions. 

For this project of mine, I need an Ultrasonic Sensor, a Gyroscope, a Buzzer, three push buttons and an 16x2 LCD. 

Can I connect all these components at once to an Arduino Uno? I mean will I run out of pins or not? Should I go for a Mega 2560 (will have to buy one though).

Edit: a GSM sim800 as well lol
I think you're fine depends on the gyroscope model and lcd. 
-For ultrasonic sensor thats 2 pins 
-the lcd needs at least 6pins but with an i2c adapter you only need 2 pins
-3 push buttons 3 pins
-one pin for the buzzer 
Plenty of room left on the uno