Recording with Mark III

I have no experience at all with recording, I just purchased an inexpensive Focusrite interface to record with, and downloaded DAW called Reaper. Is there a way to record directly through the amp? Or do I need to mic the amp?

There's a couple of ways to go about it, some more expensive than others, some more convenient, and some more flexible. In no particular order:
1) Mic your cab. This is the most inexpensive way, but also the most "limited" compared to the other two. The only drawback I can see would be if you can't turn the amp up loud enough to where it sounds like you want it to. Otherwise, great alternative, it's been used forever and I think we all agree it works.
2) Use an emulated line out device (Mesa's CabClone for example), a loadbox with a line out (not emulated, like the Suhr loadbox), or simply the Direct output on the back of your amp. This way you don't have to mic the cab. You have the choice of using the sounds your emulated line out provides (if you use one of those), or using external Impulse Responses that "recreate" the sound of a specific cab. Keep in mind that if you use the Direct out from your amp you'll have to find IRs that emulate both a speaker cabinet and some sort of power amp emulation.
3) Use something like the Two Notes Torpedo Live which is both a loadbox and a power amp/speaker emulator all in one box (the important part here is that you can choose whether you want just the power amp emulation, just the cab emulation or both). The main difference between using IRs and the Torpedo Live is that, when using IRs (unless you are able to make your own, which I'm assuming is not your case), you are "stuck" with the IRs you can get/download/buy (which is not necessarily a bad thing, as there are tons of IRs out there). However, the Torpedo Live not only emulates the speaker cab, but it also emulates different mics, and you can change the position of the mic respective to the cab in real time like you would in real life.
Hope that helps, cheers!

I bought a Pyle PDMIC78 (cheap clone of an SM57) on Amazon that I use to mic my cabs and record right into a USB port into Audacity. It works well enough for me to get sound into a file and only cost about $15. Sooner or later I'll get a real SM57 but this was a nice easy way to learn to record.

Micing up the amp sounds the best I think. 1 Sm57 aimed a bit of the center cone and 1 MLX ribbon beside it and right up next to the grill cloth.

Both of these mics are inexpensive but for a good tone they do the job. Royer makes way better ribbon mics but for a home studio, these will work. 57 gives you the punch and snap and the ribbon is the warmth. Mix as needed.

You can even pic the combo with the 57 and mic a ext cab with the Ribbon (Note there are some phasing things that happen but you can move things around to help null this).

GL I use a Focusrite Saffire Pro 40 and I really like their gear!