I think you could get as many different answers about preferred formats as there are users here!

I guess the real answer is that you need to choose your format according to the target of your output. If burning a DVD, for instance, you have no alternative but to use mpeg-2. If it's a Blu-Ray, you can choose between Blu-Ray compatible mpeg-2 and AVCHD. The latter is, of course, mpeg-4 but rendered using the H.264 codec. It is much more compressed that its mpeg-2 equivalent.
MPEG-4 and AVCHD are seemingly the formats
du jour at the moment. But you also mention AVI and MOV. In a sense, by doing so you are missing the point that these are not really formats in their own right. They are really only wrappers which can enclose a fairly large variety of of formats.
AVI, for instance, can run from totally uncompressed (or 'true') .avi which is huge -- running to around 65GB per hour. But it can go through an enormous range of compression to highly compressed ones at the other end of the spectrum, including mpeg-4. Much the same could be said for MOV. However, there is a bit of a cloud over the head of MOV since some wormholes have apparently been discovered in some of its associated software which make it a security risk.
There are also other highly compressed formats which are increasingly popular, such as Matroska or .mkv -- though that too is really a wrapper format. An emerging format is H.265 though you need fairly high-end hardware to run that.
All in all, therefore, I think your would be safest with mpeg-4/AVCHD. Places like YouTube can run it; AVCHD is a Blu-Ray format; and most recent HDTVs can play it either directly or via a USB thumb drive or external hard drive plugged into a Blu-Ray play attached to the TV.