Should I get a 1TB SSD or upgrade my processor?

Get the SSD. It directly addresses the slow loading and should make installing and updating games feel much less sluggish. Put Windows on it too if you're currently booting from the hard drive.

It won't automatically give you more frames per second, but the i5-8400 is still a reasonable match for a GTX 1650. You can revisit the processor later if a particular game is struggling.

Check your motherboard before ordering. If it supports an NVMe drive in an M.2 slot, that's an option. If not, a SATA SSD is still a huge step up from a hard drive. You don't need the fastest drive on the market for this upgrade.

If 1TB stretches the budget, a 500GB SSD could still hold Windows and some games. Keep the hard drive for larger files or games you don't mind loading more slowly. For important files, keep a separate backup too.

For gaming performance, check what's limiting you before spending money on a CPU. If the GPU is consistently close to fully utilized while a game is running poorly, a faster processor may not help much. CPU-heavy games can be different, though, so it depends on what you play.

Also price the whole processor upgrade, not just the chip. Depending on what you choose, you might need a different motherboard and possibly memory as well. For the loading and update problems you described, the SSD is the more direct fix.