1) In the first day try to do one of the following, by the end of the first night try and do the other:
(a) Get down a dungeon as soon as one shows up or burn down a monastery to get an artifact.
(b) Conquer a Mage Tower and get at least a couple of spells.
2) Skills that give you mana tokens or crystals are really useful. Stopping at mines is also a good idea i.e. it's hard to conquer a city without a good stock of crystals.
3) You need to get a core tile out before the end of the second night so you can recruit elite units in Day 3.
4) The first city should be conquered before the end of Day 3 at the latest.
For #1: You should always be very happy to see a dungeon. Mage towers are also good, though you'll probably take wounds conquering them. But I wouldn't typically burn a monastery too early in solo play. Not only is the reputation hit annoying, but ideally you want them to keep generating advanced actions for you to buy.
I definitely agree with #2, it's easy to underestimate the power of crystals.
#3 and #4, I suppose, but I would express the principle as "don't dawdle". Beating a city night 2 or day 3 is generally desirable, but really you want to do it whenever you can get your hand to be suitable - I had one game where Tovak somehow took out two gigantic cities both on night 3. (I think day 3 was spent clearing the surroundings and then detouring back to a mage tower to buy Tremor, without which he wouldn't have had a chance against the larger one.)
I would add that the better you get at the game, the easier you'll find it to arrange your hand to meet whatever challenges come your way, rather than trying to find particular location types to beat. Don't underestimate the value of killing orcs for fame, either.