These are some basic pieces of equipment that are relatively inexpensive and can allow you to successfully work on your fitness from home:
Stability ball (the big one)
Resistance bands (both the kind with handles and the loop ones for monster walks)
Mat
Sliders
Bands w/paddles
Any kind of weights! Dumbbells are preferable over barbells because they’re more versatile.
These next few things are not necessary but can add an extra element to home workouts. They require a bigger investment:
TRX straps
Balance Disks
Bosu Ball
body bars
Bench
Plyo box for jumping