Okay, as Jorpho says, the concepts remain the same, but obviously times change and utilities get replaced by newer and better things. If you check out one of my translations, the readme will tell you everything I used to do it, so let's copy-paste from my template.
FCEUX (best NES emulator for debugging and hacking)
MEKA (the best Master System emulator for debugging)
Mednafen (very versatile debugging emulator)
Bizhawk (for certain debugging tasks)
HxD (general hex editing)
Tile Molester (graphics)
FDSExplorer (examining FDS file structure)
WindHex32 EX (checking/touching up text)
Pointer Tables (dumping/inserting text)
DTE-Opt (making DTE table)
GIMP (graphics drawing and editing)
This is the extent of the tools I've used for my releases. HxD is the smallest, simplest hex editor around which can do virtually everything you need when hacking files, aside from a lack of Japanese text support (which apparently the author is thinking of adding), but for that you have WindHex32 EX. I use Windows 10, so everything you see here works fine under that.
With HxD, WH32EX, Tile Molester, and a good emulator for your chosen platform, you can achieve great things.
You still need to figure out how to hack, of course.