My taste in video games has changed significantly over the years. My brother and I were never allowed a games console when we were young, but a compromise was the Game Boy, from the age of about 10 or so. Most of out early gaming experiences in the meantime were on friends' consoles or computers at their houses. We both loved playing Zelda: A Link to the Past on our cousins' SNES, unfortunately they lived so far away we only saw them every other year or so.
We enjoyed Zelda so much that we shared a copy of Zelda: Link's Awakening for the Game Boy, and saved up our pocket money to buy a second-hand SNES and a few games from a friend-of-a-friend at school, with grudging acceptance by our parents. It was wired up to a tiny black-and-white TV in my brother's bedroom. This was followed by the joy of finding a copy of Z:ALTP in a second-hand games shop in Bath on a family holiday, for £7. Oh, what happy days!
A few years later we bought a second-and Nintendo 64 from another friend, with the next Zelda game (Oracle of Time). Among the cartridges he sold us were Goldeneye and Perfect Dark, which with the addition of extra controllers, made for a fair bit of fun with friends.
A family computer was bought as I entered secondary school. As a teenager, I asked for a second-hand computer of my own. It wasn't much to write home about (a 486 with 32Mb RAM, £95 in 1998/99) but as it wasn't the family computer, I could muck about with it, tweak it, break it, and take my time to fix it without other members of the family breathing down my neck.
The Computer Technician at high school started an impromptu club, whereby a special 'games' login was enabled after school hours, with Doom (and later Quake) installed and available for us to play over the school network. I met a number of fellow geeks, and after the Technician was laid off due to a variety of budgetary issues, we started dragging our own PCs to each others' houses on weekends for sleepovers and network gaming sessions. (We'd never heard the term 'LAN party'.) That lowly 486 saw a number of upgrades, and whenever one member of the gang saved up enough to buy a new X for their gaming PC, the old one would be offered to a friend, who'd have a spare to pass on, in our own version of a rolling stock cascade.
PC gaming was thus my main focus for nigh on a decade, playing a wide variety of shoot-em-ups (Quake, Wolfenstein, Battlefield 1942, Call of Duty (1!)), strategy games (Total Annihilation, Supremacy, Galactic Battlegrounds), simulation games (X-Wing vs Tie Fighter, Silent Hunter), and many many rounds of Worms. I bought a second-hand Gamecube to keep up with the Zelda games, as well as keeping pace with Nintendo's handhelds. As the group split up and moved away to Universities etc, the opportunities to play PC games together diminished. I never got in to online-gaming, particularly with strangers: much of the attraction was to share a room with friends and talk in person as we played against each other. My 'gaming PC' hasn't seen any upgrades in almost a decade now, as I can't justify the expense, and I've swung back multiplayer games on consoles in split-screen with friends. I hardly play shoot-em-ups any more these days, perhaps because I still feel that the proper interface for them is a keyboard and mouse, not a gamepad. (I will make allowances for a few console-based titles, such as Perfect Dark and Portal.)
My wife (whom I met at university) likewise wasn't allowed a games console when she was young, mainly due to the prohibitive price tag. When she moved away to University, she bought herself a second-hand Mega Drive, in order to play the Sonic games that she so loves. When the likes of Gamestation clearing out old games and consoles that were sat in their stockrooms, she cleaned up, and ended up with a collection of at least 50 Mega Drive titles, as well as a Saturn and Dreamcast. (The Dreamcast's ChuChu Rocket is a fantastic party game, and has been re-released on some mobile platform, IIRC.) I also introduced her (and several members of her family) to the Zelda franchise, something we now enjoy together, despite it being a largely single player franchise.
Along the years, we haven't parted with any of the above consoles, so with recent acquisitions, we have eight different games consoles wired up to our TV at the moment. There's also a collection of hand-helds, including five different generations of Game Boy / DS, and a Sega Game Gear. The key thing that's made the collection affordable is the fact that they were all bough second hand (or, in the case of one handheld, gifted new). Having said that, the Nintendo Switch has interested us enough the we might consider buying it new when it's released...
. I like PC gaming, partly because you can disguise the gaming part of the PC and tell your parents it's for homework! I agree with you about buying your own PC purely for not getting the blame for its faults. I nearly downloaded a virus on ours, and boy, I got murdered that weekend! 
