To elaborate on things a bit/act as a "what I would want someone to tell me".
MS does not like hacked consoles on xbox live (cheats
, them not being paid for games and all that). With the modded DVD drive stuff (which this may not have) that was a bit harder, but a modded dashboard lights up like a Christmas tree and they have a habit of banning both the console and profile. There are stealth services that allow some people to play some hacked games but they are costly and not really worth it.
Some RGH machines will have a dual NAND and can then swap back to being a normal console. This is a nightmare hack to keep sorted as you have to transfer files around, make sure things are clean, maybe have dual installs of things so one can be hacked, and I would suggest you get a second legit/DVD modded console here instead if online is a thing you want -- they go for next to nothing really, indeed a year of xbox live (MS actually managed to convince people to pay for online play) is probably going to cost more if you buy a console without controllers (presumably you have some to use from the one you already have).
Personally I go one further and keep any legit profiles far away from my hacked consoles and anything that touched a hacked console is never going back on a legit box I care about being banned. Now I don't care if I am banned (never going to pay for live myself) but would rather not trouble friends that do so play that as you will.
People on RGH machines will have three primary methods of playing games
1) Internal hard drive.
Regardless of what you end up doing I would suggest checking your games/ISOs with a tool called ABGX360.
Most hacked 360s were probably this, and technically even on a stock drive you can burn a DVD full of stock arcade games and play the demo versions.
Anyway the DVD drive performs various checks on the DVD it is running. This means if you hacked the firmware of the DVD drive it would play copied discs, a massive security oversight but never the less one that was made and used extensively.
In and of itself RGH hacks do not mean you have a hacked DVD, however it does mean you can download the DVD keys (each drive has its own key) trivially in software and I believe there are flashing tools to flash custom firmwares to them these days (as opposed to needing custom setups). Not sure what goes for your setup offhand and frankly unless you/a friend has a large collection of burned discs and no copies of the ISO on PC (or desire to rip it back) I would not bother.
The internal (and technically USB) might well make use of something called GOD files aka games on demand (basically what MS calls their downloadable games on the 360). When you copy a 360 game to the hard drive on a normal one and use the disc to make sure you still have the game but run from the faster hard drive that tends to be known as NXE, this is but a few bytes (none of which is game data) from GOD.
This is why you then have tools to convert from ISO files as downloaded on most of the internet to GOD files for use on the internal drive (not sure what the kids are using right now but ISO2GOD I always liked), and those tools to convert NXE installs to it (See NXE2GOD).
It is usually also far easier to install XBLA (xbox live arcade) titles and DLC to the internal hard drive rather than USB, but there are various methods.
Bonus for JTAG/RGH family hacks is while a stock 360 is somewhat limited in the options for internal drives (it needs to be a given model and you need to flash security sectors to it) then hacked ones can use whatever they can fit in there.
Depending upon where you find your XBLA and how you launch it you may have to crack it (a trivial thing -- each XBLA "demo" is the full game but needed a tiny tweak to have it appear as full, there were however some older methods to allow people to borrow profiles and use things on those profiles so some XBLA and DLC was made for that).
USB. Two approaches here.
1) Towards the end of the 360 life MS allowed people to plug in USB drives and take up some space on them to have them appear like a memory card/small hard drive. XTAF (FATX backwards, FAT being Microsoft's older file system and FATX first being seen on the original Xbox) is the name of this one. Most only use this if their 360 lacked an internal hard drive or to keep a profile on it to go round a friend's house with.
2) Plain old USB. Here people will typically extract an iso to folders on it and run it from there. This is harder to run from the stock menu so you will use other tools/loaders, if you are not running the stock menu or autobooting into a hacked dash then most of the hacked ones will have various directories they monitor and allow you to select/boot games accordingly.
Getting stuff on there.
As something of a hangover from the original xbox when the 360 got hacked most went back to FTP. FTP = file transfer protocol, a very old but surprisingly robust means of transferring data over network. Join your network, log on via a FTP client (filezilla is what most would suggest) and transfer away
It is not surprising at all to see people fire up the 360 with a USB drive still plugged into it and transfer files over the network to it, even if they could theoretically plug it into their PC and copy files across like that. Might be slightly quicker these days if you are using a USB 3.0 drive as the 360 is USB2.0 (about 30 megs a second on a good day).
Also means you can transfer from the 360 drive to the PC if you did want to back things up for later use, or maybe some saves for editing saves.
The file types. I already mentioned GOD and related stuff above.
The more general types though are the STFS file format.
This is broken down into three sub formats
1) LIVE. This is what most XBLA, DLC and the like comes as. Without suitable permissions this will not run, fortunately RGH machines don't care about such things.
2) PIRS. This is a rarely used format but some DLC from things like game of the year discs comes as this. You can install this just fine on even a stock machine if you wanted (indeed the USB-XTAF stuff was often employed here for those on stock consoles wanting things like the first two borderlands DLC packs).
3) CON. Mostly for save files, profiles and the like. If you are editing these on a PC this will be what you are editing. Technically there were some old hacks that allowed you to convert from LIVE to CON and have a game accept it (mostly music games as their DLC was often basically a MP3 and a short note list which is less than some save files for things like an open world game).
There are also title updates (TUs). These are much like they sound and are the updates for games to fix bugs, add new features and whatever else. If you want what the update gives you then go find it (there are large libraries of them around the internet, often free and easy to download) and install it accordingly. Some people want ones with various bugs, and things like fable might have a special themed day you might want to experience.
Save editors are on a game by game basis really, and most of the time profile editing is silly but as you are only really going to get banned for it and the system is not bothered about being able to connect to MS services then edit if you want. Be careful as a broken profile is not fun to get back without it (basically take backups).
Le Fluffie is a tool I like to have for playing with 360 stuff like this.