Dom - 2E0WHQ
Strictly speaking, being the 'FM'AC, then traditionally antennas would usually be vertically polarized for FM, however, as we know, as the FMAC precedes the UKAC (SSB), most operators will just put up a single antenna (beam) if they are going to take part in both. For 2m, I use a simple Alton Moxon antenna, which does have some directional properties, but it's beam is wide enough to allow me to pick up stations which I'm not pointing directly at. A good Omnidirectional, horizontally polarized antenna is the Wimo 2m Big Wheel. That seems to give consistently good results without having to go to the effort of a beam.
For 70cm, I use a Diamond A430S10R2, which is a 10-element Yagi that gives fantastic results. It's a nice compact beam and if you want to make it smaller for packing away, then you can just undo the element wing nuts and turn the elements sideways so the antenna takes up less space.
For any beam, I would probably recommend a centre supported beam, rather than a rear-end supported beam as it puts less stress on the rotator, as the load is balanced equally.
For 2m antennas, take a look at what Diamond, EAntennas and Powabeam have to offer. It really comes down to a personal choice and an antenna which meets your size requirements. At the end of the day, all Yagi antennas are pretty much working off the same principles, Reflector, Driven Element and Directors, there is nothing mythical about them really. The more directors you have, the greater the gain, but the narrower the beam.
As for rotators, there are some 12v ones out there, but not many. There is one specific 12v supplier, whose name escapes me for the moment, but they are very expensive if I remember correctly. From a recent post in one of the Facebook contesting groups, it showed that most people either used a Leisure Battery with a 12/240V inverter and just used a mains powered rotator, or they used a generator to give them 240v for the rotator. I personally use a leisure battery and an inverter.
Welcome to contesting! I'm sure you'll get lots of varied opinions and suggestions.