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I do not have a synthesizer and I was wondering if anyone has a good program they know of that can both play and record sounds off a digital keyboard, some sort of application where you can both record and play different virtual instrument for Windows 10.

If you are looking for a DAW (Digital Audio Workstation) that can host VSTis (Virtual Studio Instruments) there are a lot options.
https://en.wikipedia.org/wiki/Digital_audio_workstation

I mainly use Cubase and Reaper.
Reaper is very affordable. http://reaper.fm/
Ardour is also worth a look. http://ardour.org/

Most popular Digital Audio Workstations are functionally very similar.
Differences are mostly found in the workflow and general design philosophy of the software.

There are tons of VSTi synthesizers out there.

If you want classic analogue synthesizer sounds the free U-He VSTis Tyrell and Bazille might be a good start.

https://www.u-he.com/cms/tyrelln6
https://www.u-he.com/cms/bazille

But if you're searching for some other instrument.. It's probably available as a VSTi emulation.

Best wishes,
cabled_mess

For acoustic instruments, amongst others:

Native Instruments Kontakt 5 Player
https://www.native-instruments.com/de/products/komplete/samplers/kontakt-5-player/free-download/

Magix Independence:
http://www.magix-audio.com/us/independence-free-sampler-software/

Beat Zampler + free SFZ-librarys or some other software capable of reading SFZ:
http://www.zampler.de/