Does it matter whether I wind the primary or the secondary first? I imagine with the primary inside the secondary you'll have less leakage. Is that right? Other reasons to put the primary or secondary first?
It very much depends on the core geometry. With 'classic' E/C-cores you will indeed have more leakage if you wind primary first, but this effect is really very small with any sane transformer.
The main consideration when winding cores is safety and thermals. You want to wind in such a way that you maximize insulation. Also, you want to put the highest heat dissipating windings on the outside. This both usually leads to primary on the inside and secondary on the outside, because the secondary will have n less windings and consequently n times as high current (in an ideal world), so as dissipation scales with \$n^2\$, your dissipation in the lower turns count will be n times as high - of course when using the same gauge wire. Even though nobody in their right mind uses the same gauge, for most practical applications it's not possible to use sufficiently thicker (let alone sufficiently fine multi-strand wire) to completely negate the dissipation scaling.
As for the insulation argument: if you're winding your own transformers, you will see that the first couple of layers will be extremely neat, but as you continue your winding plane will become messy. On the outside it will be fairly uneven. This impacts how well you can guarantee insulation distance, even when using thick insulation tape. This is why it is better to put your primary on the inside: that way you have the bobbin on one side of your primary (which is a perfect working surface and is already well insulated) and clean windings on the other, which you can tape up to be nicely electrically isolated, without the risk of uneven wires poking through on the edge of your winding window.
Of course, you can also elect to use copper tape for the secondary, which negates much of what I have said here. However, copper tape is not the perfect solution: it usually comes unterminated, so you need to solder on wires to connect it to your outside connections, which is an annoying irregularity in the winding window. Consider how you do this before you start winding, e.g. you can fold the copper tape over the wire and then solder it to make the connection less uneven.