We've all been there: you've played a bunch of records before without any issue, and you start to play a new record when you realize that when it gets to a particular part of the song, it skips ahead. You clean the record with a brush and you clean the stylus and nothing changes. It's never happened with any of your other albums before, so obviously it must be this record's fault, right?
It's not as likely as you think.
Cheap record players fail to properly play and sometimes even damage records due to several factors, primarily related to lower quality components and inadequate design.
These issues include:
Missing Tracking Force Adjustments
A properly set tracking force is crucial for minimizing wear. Cheap turntables forgo mechanisms to make adjustments for precise alignment, and often leads to too-low forces that allow the stylus to fall out of grooves or too-high forces that dig into grooves. This is why you'll sometimes find inexperienced collectors attaching weights or coins to the top of their tonearms in an attempt to prevent skipping.
Poor Stylus Quality
Many inexpensive turntables use ruby or sapphire styluses, which wear out faster than diamond styluses and can damage the record grooves as they wear down. The most commonly found stylus, identifiable by its red plastic appearance, is made of sapphire.
Anti-skate (Or Lack Thereof)
The anti-skate is designed to stop the tonearm from drifting inwards and damaging the record. Many cheap turntables lack this feature or have poorly designed systems, resulting in uneven groove wear.
Internal & External Vibrations
Cheap turntables may be more susceptible to vibrations, both from the motor and from external sources such as conveniently built-in speakers, which can cause the stylus to skip or damage the record. This is also sometimes attributed to placing speakers near or on the same surfaces as turntables without proper shock absorbtion.
So TL;DR: cheap turntables will only give you your money's worth. As you collect more records–and presumably rarer and more treasured ones, you'll want to properly take care of them.
Consider upgrading to more advanced models as you see fit. The most common 'upgraded' beginner's turntable is the Audio Technica LP60X, which should get you past the most common of skips and provide a vastly improved listening experience. That being said, even the AT-LP60X isn't immune to hotly-cut records with wide grooves that might cause the stylus to jump once in a blue moon.
If you need somewhere to start on your journey to a new turntable:
Here's a list of the most common and reliable turntables found today