The console you want is the Wii U.
Lets look at price first. I've taken these prices from Amazon UK as that is where I do most of my shopping and they usually have good prices.
An Xbox One and Call of Duty Ghosts will set you back a total of £477.99. That's £429.99 for the console and £48.00 for the game. Note that I selected that game only because it is available on Xbox One, PS4 and Wii U to help illustrate my point.
So what can £477.99 (or less) get you if you want to go for the Wii U?
Wii U Premium Pack - £239.99 (includes Lego City Undercover and Rayman Legends)
Call of Duty Ghosts - £44.99
Super Mario 3D World - £39.99
Pikmin 3 - £31.99
Zombi U - £15.99
New Super Mario Bros U - £39.99
Batman Arkham Origins - £22.49
Need For Speed Most Wanted U - £19.95
Deus Ex - £16.99
Total - £472.37
I've selected the games listed above partly at random, but they are also among the best games available for the console and includes the obligatory CoD: Ghosts.
Turning our attention to PS4, you are looking at a total of £396.14. That's £349.99 for the PS4 and Call of Duty Ghosts will set you back £46.15.
If we drop the two Mario games from our list above, we get the price down to £392.39.
Xbox One and a single game OR a Wii U with TEN games.
PS4 and a single game OR a Wii U with EIGHT games.
And let's not forget that if you already have a Wii at home you can use all your existing controllers (and other peripherals) on your Wii U so you might already have everything you need for multiplayer action, rather than shelling out £49.00 for a second PS4 pad or £44.00 for a second Xbox One pad.
We should also note that if you get tired of the eight or ten games you get with your Wii U and you already have a Wii you can play all your old games on your Wii U. Try loading up one of your PS3/Xbox 360 games on their next gen replacements and you're not going to get very far!
So to reiterate the central message of this post, forget PS4, forget Xbox One. If you want a new console AND some awesome games, get a Wii U.