Either laptop or desktop?

Desktop is by far easier but laptop is convenient. I would go with laptop

I don't know if you're in the UK or the US, but you could try searching for an intel i3 or i5 with nvidia 740m graphics & at least 4gb RAM. They in around the 430 to 600 pounds mark in the UK.
If you can get an 840m, even better.
You can play Minecraft on full settings & more demanding games on medium using that.
You'd probably pay a similar price for a desktop, which would be better for serious gaming, but the laptop is portable & meets your requirements.

Although laptops are portable… They are inherently short lived for gaming. Laptops are not upgradeable You are stuck with what you have, where as a desktop can retain its usefullness through many additional years by upgrading the parts you need to to keep up with the demands of the games. Ultimately it is you that needs to decide what is more important… Portability or longevity.

Laptops have everything that a desktop has minus the ability to change parts. A desktop is also portable as in all you really need to move is the tower and the cables needed to hook into a television or monitor. A desktop can be upgraded as needed and tends to last longer. It really depends on your needs. Do you constantly need a computer on hand or can you access your computer at any given time?

Laptop = convenience
Desktop = Compatibility and upgrades.