It is hard to look past Honda, but don't rule out Suzuki or Yamaha. They're all quality Japanese items. There's a lot of others on the market and a lot coming on to the market from makes I've never heard of before or from makers known for other things. They'll do a job I guess, but feedback on them is sparse. They tend to be a fair bit cheaper than the above makes, but there'll be a reason for that.
I'd rather buy a second hand Honda, Suzuki or Yamaha than a new whatever. As for size, they differ a bit, so take a look at a few to see which suits you and your purpose. As for engine size, I'd say for most purposes, 350cc is a minimum and 450/500 is better and does pretty much everything, although you can get bigger engines.
I don't know whether you need to have a road registered one, but if you do, there's either a PLG registration, taxed as a car, needs insurance and MOT, or the agricultural registered ones, still need insurance for road use and the tax is free, no MOT required. But you're limited to 1.5 km on the road for each journey.