I currently ride a Cannondale Catalyst 3 and I've noticed my chain doesn't like to catch on my front chainrings. I'm very new to mountain biking and I am on a very low budget, so is there any way to fix this problem without spending $100? Do I have to get new chain rings, a new crankset, or something else? I'm a student and I don't have all those fancy tools that are specially designed for bike work. Any suggestions on what I should do?
Chains stretch over time and, as they do that, they wear the chain rings and cogs to match their new size. By the time the chain starts skipping on the chain rings, the teeth have become so badly worn that there's no alternative to replacing the chain rings, chain and rear cogs. Unfortunately, that probably will cost $75+ in parts.
The way to avoid this in future is to replace your chain before it's stretched so much that it destroys the rest of your drivetrain. Chains themselves aren't very expensive and replacing them is one of the running costs of a bike. You can check how worn your chain is by using a chain checker – you can get them more cheaply than the one I've linked to.
Based on reading your question and the comments, I feel the combination of worn chainrings coupled with the new chain and cassette is causing your chain slip. The Cannondale Catalyst 3 sports a Prowheel triple crankset with 44/32/22 tooth rings. This is not Shimano brand but does share the same 104/64 mm BCD as Shimano rings of similar type. Amazon lists a Prowheel crankset as you describe for around $68. Pretty reasonable as far as cranksets go, and I see some listings on a Google search for even less. It wouldn't be unreasonable to expect individual chainrings to run between $10-20, especially for the middle and small ring. A 5 or 6 mm hex wrench (aka: hex key) and standard screwdriver are tools needed for chainring replacement. If your system has an external bottom bracket, the cranks can be removed with a 5mm hex key at the pinch bolts of the left crank arm, and the preload bolt in the center can be engaged with a 10mm hex key or similarly large torx bit. This is NOT ideal as one is supposed to use a special tool that properly engages the splines of the bolt. This is also typically plastic and can be easily damaged. However, there is not a great deal of torque required on this bolt so there's a bit of leeway there when one doesn't have specific "bike tools."