I don't have an original monitor but I was hoping there was a cheap and reliable method of connecting a VGA monitor...
A cheap ISA VGA card off ebay will get you going. Since you have a 16-bit ISA bus, you have a lot more choices, so that's good.
What, if anything, should I check preemptively to clean/replace so I don't damage the computer when it's turned on? I know old caps can go bad for instance, but I'm no electrical engineer.
If everything looks good, I'd just plug it in and try to power it on. Be ready to flip it off again if something smokes. If nothing smokes, and you happen to have a multimeter, check the PSU voltages (it will probably need to be plugged into the mobo to provide enough of a load for the PSU to work). But if you don't have a multimeter, don't worry about it.
(Re-edit: I stand corrected by maxtherabbit re: parking MFM drives. Nevermind about all that. ;P )
Electrolytic caps fail open, so they generally don't break anything. The machine just may not work until you replace them, if they are bad, or may work but behave weirdly.
I think some of the IBM expansion cards of that vintage used tantalum capacitors across power rails, which will fail closed and keep the PSU from starting up, but I have never heard of them actually breaking anything. They just have to be replaced.
Is it possible to get networking (Ethernet) on this PC?
Yeah. Again, since you have a 16 bit ISA bus, you have a lot more options. You'll have to provide what is called a "packet driver" specific to your network card to make the card work, though. Some programs may require a TSR TCP/IP driver, too. TCP/IP under DOS is kind of a mess to get working, and I am definitely not an expert on the subject, but here is a page that goes over it:
https://www.brutman.com/Dos_Networking/
I've had good success with the 3c509 network cards in ye olden days, although I do not currently own any. Beware of jumperless "ISAPNP" network cards, as they require configuration with ISAPNP software that will only run on a 386 or better, IIRC. You probably want an older card that actually has physical jumpers on it.
While there are some DOS web browsers, they're behind the times vis-a-vis HTML specs, and I wouldn't think a 286 would run SSH very well. You can still use things like telnet, pop/imap, ftp, gopher, etc, though. I find NFS extremely useful for file transfer, although there are probably other things that are easier to get working if you don't have Unix machines that also need love. And the PC NFS driver takes a crapton of memory.
I have a card with two RJ-11 outs and I'm guessing this is a modem card?
Probably. It's basically junk, IMO. All the BBSes nowadays are on the internet with a TCP connection, so once you get your network card going you can just telnet to them.
If you got adventurous, you might be able to cut some traces on it somewhere and reroute a TTL-level serial signal to a Raspberry Pi or something and make it into a wifi modem, but if you are going to get a network card anyway then.......
Will I need floppy disks to initialize the computer and/or the expansion cards? I have two RAM expansion cards and two different graphics cards and the modem(?) card.
I dunno. It's liable to have an old MFM hard drive in it, but who knows if it still works after all this time. If you can post some pics, we can better help you ascertain what you have and how to get it going. With a VGA card, a hard drive, and a soundblaster, a 286 will play a lot of middle-era DOS games pretty well. I recommend Dune, once you get the machine running. :D
Welcome to here, btw. I think this is one of the best communities on the internet, and I hope that your time here will be as enjoyable as mine has been.