This is an interesting time for everyone but really, there won't be a better time to start a little home "gym."
You don't need a lot to get a good workout in, in fact, you really don't need anything but some equipment can help to add variety, resistance and progression.
Low Budget/Start Here:
A set of resistance bands with handles: Grab a set of resistance bands with attachable handles. Generally speaking, a good set of bands will cost you around $25-30 but they will allow you to do ALL THE THINGS. Bands can be a suitable replacement for dumbbells, and depending on the strength you need, can give you enough weight to get a great workout in. You can attach them to different places in the house, to each other, or use in a variety of positions. In addition, bands are small--they work great for travel--and can easily be hung in your living room or stashed in a basket somewhere.
We like this set linked here*
Mini Bands: Mini bands cost around $8-$15 so spend 2 days worth of coffee on some fitness gear and grab a set. Mini bands are a great option for adding resistance to lower body exercises and can also be used for some upper body strengthening as well. They aren't as versatile as a set of bands with handles, however, for items so cheap and small, they can get a decent amount done.
Yoga Mat: Grab yourself a mat for some more comfort during floor exercises. While you definitely don't need one, it will help in preventing rubbing/bruising during core exercises and movements where you're on your back.
Pretty much any mat will do. We buy this one.
Sliders: These little discs look like fun but they will bring some challenge! Throw some sliders in for core work and even some leg exercises. Again, any will do but we like these in our home and studio.
Medium Budget/Middle Ground
A pair or two of light to medium dumbbells: You don't have to go crazy and stock an arsenal of dumbbells--one or two pairs can get the job done. Select a pair of light-medium weight dumbbells that you can use for a variety of exercises. Dumbbells range in price but should be around $1 per pound. We like hex dumbbells, meaning metal grip with hexagon ends but, it truly doesn't matter.
Try the ones from Amazon.
Suspension Trainer: This might even fall into the category of low budget depending on what kind you get. A suspension trainer is a pair of straps that you can either attach to a wall anchor (in a stud) or rig up in a door that allows you to do a variety of pulling, pushing and core exercises.
For most of us, sticking around a low to medium budget is plenty! Adding a few more dumbbells (or a pair of adjustables) will be enough to get pretty much any job done!
If you have a budget to play with, you may consider grabbing a piece of cardio equipment like a stationary bike or treadmill to get some cardiovascular exercise done inside.
For more advanced lifters, you may want to consider a rack + barbell + plates for squatting, benching and pressing and having a rig to attach things to but it's definitely not necessary!
Have fun setting up your home gym!