Australia: NZ nationals can travel to Australia without a visa, indefinitely I think. I arrived at Brisbane port and passport officials came onto the boat and checked my passport before I could disembark. The German couple also on board seemed to get through without any trouble.
Indonesia: a one month tourist visa on arrival. I read that these can be got on arrival at ports but this was the one country I arrived into by air.
Malaysia: ninety-day visa on arrival. I got this at the port of Melaka and I've read that they can be got at other ports such as Penang.
Thailand: two-week visa on arrival. This is different from if you arrive by air - in that case, the standard visa is for one month but arriving by land you only get a two-week visa, for what reason I don't know.
Cambodia: ninety-day visa on arrival. There are some scams at the border where officials will take you to an office some way from the border and while they give you a valid visa, you end up paying more than you have to. That's what happened to me but I suspect that if I'd just ignored those officials and gone right to the border I would have got it fine there.
Vietnam: one-month visa in advance. They might be possible to get when arriving at the border but the standard practice seems to be to get it in advance. I got mine while in Phnom Penh, my hotel getting it for me for a reasonable fee and it only took one night between my requesting/ paying for it and it arriving.
China: one-month visa in advance. Again, the standard seems to be not to get these on arrival. There seemed to be plenty of tourist operators and backpacker places in Hanoi that will get them for you for a reasonable fee but they take a long time: mine took about a week.
Mongolia: two-week visa on arrival. Even though I got mine on arrival, I recommend getting it in advance (which is possible). In advance seems to be the standard practice and wanting it on arrival met with surprised officials and negotiations over fees and visa duration.
Russia: one-month visa in advance. This was by far the most difficult visa I had to get. In addition to the application form, you need a detailed itinerary and an invitation from a Russian tour agency. There are tour agencies with websites that will provide these for a fee. I was fortunate in that Kristen already had got these for me when she was getting her's in New Zealand. In China, the Russian consulates in Beijing and Shanghai will not give visas to those who are in China on visas of less than 90 days, i.e. you have to be a Chinese national or living and working there. The Russian consulate in Hong Kong, however, is more flexible and will give visas to travelers so go there or even do what I did and send your passport and forms to someone there who will do it for you (the HK consulate doesn't seem to have a problem with not being there in person) - but remember you need your passport to check into any hotel in China so only send it if you've already checked into a hotel for at least a few days (or alternatively sneak into your girlfriend's room when the staff aren't looking, until your passport comes back!)
Finland: no visa needed. I'm not quite sure what the situation here is. They stamped my passport and there doesn't seem to be any date-limit and no fee to pay. So I think New Zealanders can stay as long as they like so long as they don't work.
The Rest of EU: there were no further passport checks through Estonia, Latvia, Lithuania, Poland, Slovakia, Austria, Czech Republic, Germany, or Netherlands. There was a passport check when I arrived in the UK which surprised me - they are in the EU aren't they? - but there I used my UK passport. I'm not sure if this is going to result in any confusion when I leave but I can't see why it would, which is not to say it won't.