I have used a dremel for a handful of mods but I would not go out buying one straight away. Maybe if you are really into case modding but for there I would probably suggest small files, a drill and a tapered reamer (it cleans up holes and makes them just that little bit bigger (or a lot bigger if you want to stay there for a while) before a dremel comes in -- they are quite crude tools, not without their use but if you are doing any kind of proper finish then not so much. I run extra wires, LEDs, put in new panels and whatnot into things all the time and a dremel is only ever there because I don't want to bring two drills or when they have some nice wire brushes to get right into tough areas, occasionally I might play with friction discs if I am being lazy and diamond burr stuff is not so bad. Now if you had an air line rotary tool we might be able to have a conversation, only just but we could have one.
Soldering irons I am less sure of -- apparently I lucked out with my cheap gear where some of the cheaper ones I used later in life were horrible and I am also not sure what goes in the US market.
Still
Equally these days you might want hot air. You can buy a fancy one if you like but aoyue stuff works for most things I need in electronics repair. Depending upon what you are doing you might also want a desoldering iron -- personally I am happy with a solder sucker and wick for what I do but if you are removing cans or ports that can hold a lot of heat you might want something with a bit more grunt.
"move or remove internal pieces" and "any guides to removing things"
Wow OK... guides wise there is nothing general, there might be a specific guide that says snap along this line. For the most part though if you are doing something like that you will already know what is there and what is being lost or shifted to another board. You need not know absolutely everything, though that would not hurt, and often times you can tell what board (if there are multiple) is doing what and thus what you might be able to remove . There is so much to all this -- even if a board section does one thing it might provide a ground to another section, which you can then remake, you might be able to remove something but it might then want a resistor to act as a dummy load, if you are splitting something in half to get the footprint down I hope you know what matched impedance traces look like (usually they will look like squiggly traces where a straight lines looks like it will do) and it goes on.
There are also a lot of multiple layer boards these days.
If you just mean I want to stand this regulator up and this thing has some fat old capacitors but I have nice skinny ones rated the same then it is as easy as it sounds.
On wire then the classic suggestion is kynar wire, this is good stuff. Otherwise it will depend upon what you are doing -- you don't want to be stuffing 20A down a hair thin wire and you don't want to be putting a multi GHz signal down some unshielded wire you scrounged from a PC motherboard.
You say controller slots which is fun as they can both do signals that need integrity and power, possibly at the same time, depending up the system.
There are ribbon cable manufacturers that do custom pitches, crossover, joining, lengths, counts, backing, points of removal and whatnot but every one I have ever tried to deal with has been a pain, even if I don't just want one offs for some of the things I have dealt with. You can rip old stuff out of printers, laptops and whatever else you like if you want but where I often find uses for my cables with connectors box (and if nothing else it is free wire) I can not say I have met anybody that does well for keeping old flat flex ribbon cables around, unless they have happened to find something readily available that does well for another task*.
*a friend found that a particular car radio would fail because of a bad cable and that a pound shop device had a more than suitable replacement. £1 is ridiculous for a ribbon but nobody cares when it fixes their £300 radio.
Personally if you are ever at yard sales, swap meets, car boot sales... I say pick up old floppy drive and IDE cables -- all that nice ribbon cable all connected together which you can solder to whatever you like. I have fixed so much stuff where conventional ribbon cable has failed.
Replacing buttons... that is a big area. Many methods
1) Make a controller adapter. Controllers are at their heart fairly basic devices or there are nice chips that can handle them these days (if you can adapt it to play on your PC it is not much different to make it work on another console).
2) Actual swaps. There are a limited number of button switching technologies out there and most you meet in games will be simple tactile domes or membrane switches. Better options will be microswitches but not so many game consoles have anything like that here. Companies will tend to stick with a tech as well and the buttons you see on top of things tend to be able to be swapped a some level. Make holes bigger, maybe lose a locator pin, file studs off if there are any in the way.... all very crude but can get it done.
3) Faux replacements. If the buttons from the console you like are given colours, dished inwards and of a certain size then that is well within the capability of 3d printing companies these days. Figure out the underside dimensions of the button caps you want to replace and make top side ones the same underside dimensions but with the top/finger facing side be the style you wish to emulate.
4) Make your own controller. Harder for the likes of the PS1 on up but older machine controllers are quite simple when all is said and done. Or you can find that a controller might have its own nice chip but if you are up on the same things you would need for the removing stuff above then you can cannibalise an existing controller and throw your own wires from controller buttons made as you see fit to it instead.
This is all somewhat beyond the scope of a forum post really -- most people playing here already have fairly extensive electronics knowledge and/or fabrication knowledge. You can happily learn all this sort of thing online but if you are asking these sorts of questions you have a ways to go yet.
More links to videos because why not, not all videos on these channels cover fiddling with things like you describe but no small amount of them are that.
https://www.youtube.com/user/bigclivedotcom/videos
https://www.youtube.com/user/arduinoversusevil/videos
https://www.youtube.com/channel/UCDbWmfrwmzn1ZsGgrYRUxoA/videos
http://www.youtube.com/user/mikeselectricstuff
http://www.youtube.com/user/thebenheckshow
http://eevblog.com/
https://www.youtube.com/user/jjward/videos
https://www.youtube.com/user/julius256/videos
https://www.youtube.com/user/rossmanngroup/videos