Changing gears early to save gas?

i drive a dodge neon 2003. so what im doing since two weeks ago is changing gears in my car really early. i use to change once i reach 2000 rpm but now im changing it at 1500 rpm. it has saved a lot of gas but it chugs a lot while trying to get speed and sometimes i hear weird noises. is changing gears early bad for my engine? if your a mechanic or really knowledgeable with sticks please help me i dont want to break my car but i dont want to waste gas either ( my dad gives me limited gas money) p.s. im on fifth gear once i reach 30 mph. and i basiclly try my best to keep it below 1600 rpm except when im on highway.