It sounds like it needs inspected and tuned, If you cannot true a wheel and own a Derailluer alignment gauge then you cannot do it at home correctly. Seek out a local co op or a Local bike shop for service.
There are two kinds of people in the world, "Those who help themselves to people, and those who help people!"
Some of the problem that I see when people set up a geared bike is that they kind of go in the order of 1,3,2. when I get a used or new bicycle in I always start from scratch. disconnect cables, check axle and crank/bottom bracket for play, check front rings for run out, true wheels and make sure they are bottomed in lugs correctly and centered/dished proper. Now the bike is ready to set up the derailluers. Check and align the derailluer hanger. This is where the 1,3,2, comes in. if you just start by turning barrels thinking it is set but it just is not right you start looking for ways to make it right scratching your head. then ultimately tracing back to what should have been done in the first place. Proper process and order 1,2,3,
Now you can proceed with the set up. set the high limit screw, set shifter to the highest gear in the rear, screw barrel adjuster all in then back out a couple turns, connect cable making sure the housing is seated in all stops. shift down 1 gear, if it does not hit the gear then cable was attached to loose. if it does shift then back out barrel adjuster until you hear the chain clanging for the next gear then back off barrel adjuster indent by indent til the clanging sound stops. Now run up and down the gears to see how they go and set the low limit screw. That is the condensed version. for a newbie it can be a bit of trial and error til you do a few and understand what action and out come have to do with each other. But learning bad habits from the get go is never a good thing. One must have the tools to perform certain tasks properly. there are no short cuts to fine tuning a bicycle, anything else is sub standard. Its just the hard truth! In your case if you follow the correct processes and it is not performing well you can go forward with more diagnosis not backward because you have done the checks to rule them out already. you will only need to take a closer look at your chain, cluster or rear derailluer and shifter its self. If you want to be your own wrench you have to go "All in" and start collecting tools as you need them. down the long haul it will still be cheaper that paying someone else.
There are two kinds of people in the world, "Those who help themselves to people, and those who help people!"