There are three generally accepted "standard" frame rates. 25 Frames per second (or 50 interlaced fields), which is really derived from television systems in primarily the UK and Europe. 29.97 frames (or 59.94 interlaced fields) (often incorrectly referred to as 30 frames per second) which derives from the NTSC television system used primarily in the Americas and Japan (originally 30 frames per second before the advent of colour television - it was changed to 29.97 for scientific reasons too long and boring to explain here), and 24 frames per second which is derived from (professional) film. Domestic cinefilm formats would often run at 8, 12 or 16fps, but that's just a sidenote.
In the very early days of television, frame rates were based upon the mains electricity frequency of the country which is how they originated, and became the standard even after advances in technology meant the mains frequency ceased to matter.
In this digital age, frame rates are less important than they used to be. If you play a file on VLC for example, it doesn't matter whether the frame rate is 25, 250 or 2.5. It's just a number, and the software will play it back at whatever rate the file says it's supposed to be. The benefit of using a lower frame rate is that your files will be slightly smaller, but ultimately if your productions will be all your own footage, and you're playing them back on computers then shoot in whatever rate you like.
However, if you plan on using material from other sources as well, you should try and have all the frame rates matching. For example, the iphone shoots by default in 29.97fps. If you shoot all your stuff in 25 then either the 29.97 stuff will need to be transcoded to 25 or vice versa. The crap editing packages like Movie Maker will do all this for you, but the transcoding will be equally crap. To get 29.97 to 25, it simply drops frames so motion will be jerky, especially on fast moving shots.
Professional packages such as Avid, Premier or Lightworks will not transcode (although often there are workarounds for importing 24fps footage into 25fps projects - it will simply play the footage 1/25th faster which is pretty much impercievable). They won't transcode because decent transcoding requires decent code and a lot of processing as entire frames will be reconstructed using information from the previous and subsequent frames. That you pay for, and would choose a package specifically designed for that application.
To summarise though, shoot whatever frame rate most of your footage will be.
I would also suggest that if you ever plan on putting it on DVD (which will ultimately output in 25fps, unless the DVD player is set to playout in 29.97), then try and stick to 25fps too, as again, it means there will be no conversion of frame rates anywhere along the production chain.
4K would probably be overkill.
Can you explain the 25/24 fps thing to me in simple terms? Every time I see a discussion on this all hell seems to break out