Raspberry Pi (or similar) to control LED strip via commands etc?

I'm trying to achieve something which I would imagine isn't all that difficult.

Basically, I would like something like a Raspberry Pi to connect to a set of LED strips.  Then via Bluetooth or WiFi I could send a command to turn them off or on from my phone. Thats it.

I'd also like the same unit to power a little OLED or LCD screen that displays the time if possible.

Does anyone know exactly what kit i'm going to need.  THere seems to be hundreds of different ways of doing this and I've got no idea where to start.