Your typical play sequence

[Qui]

I'm rather non-linear ;)

1. SMC->UD->HMV (unless I'm playing something that I think might not survive that, in which case I'd start with VD/DD and PC - skip to 3 ;))
2. Either dive to Dwarftown to get the first quest asap (made sense in 1.1.1, not sure if does now), or go back to Terinyo first.
3. Terinyo quests - VD/DD, PC, visit Holeinthewall, maybe kill Hotzy if strong enough, maybe hang around in ID if I feel like it.
4. Dwarven quests.
Here I do things however I see fit for the char. Usually dwarf quests and clear CoC down to Khelly. Farm herbs if can/need. Train with Garth if any stats really low. Griff. Pyramid whenever I'm in the level range and feel like I can handle it, Darkforge if I have a way to deal with golems. ToTHK (may be earlier if I can't find monster for 1st quest). Proceed deeper into CoC if I find AoLS or just decide to let Khelly die (option nr 2 usually).
5. Water Temple.
6. Clear CoC down to wall of fire.
7. ToEF.
8. Dive into CoC. Now I don't explore much, just descend, unless there's something of interest (vault, shop). Explore Casino Shop, but don't gamble. Rarely did in 1.1.1, don't like Casino much. Just mass identify and grab what I can afford. Usually I go all the way down to D:48 before backtracking.
9. Temple of Air.
10. Temple of Earth.
11. Temple of Mana.
12. Close the Gate.
13. Unicorn and get out.

[gym21]

With healing:

1) UD dive
2) Get to Dwarftown, acquire portal quest and kill random monster quest. Hopefully I've found a coaligned altar before I hit DT, if so double livesac precrown.
3) Level to 16 in big room, herb farm/stat train as I go
4) Pyramid
5) Dusty dungeon for wand
6) Dwarven graveyard
7) Talk to Druid, kill Keethrax if I feel like dealing with DD:7. If not, just generate it.
8) Finish DT quests
9) Darkforge raid and hope for wishes
10) CoC dive to wall of fire
11) Water temple
12) ToHK (grind to open up Minotaur Maze if I think I can do it)
13) Minotaur maze (optional)
14) ToEF
15) Dive, kill temples as I find them
16) Keep diving if Kelly saved, Unicorn quest otherwise
17) Kill everything on D:50
18) Wander off.

I almost never do the Rift or the Puppy Cave, and I mostly forget to visit Holeinthewall. Without Healing I generate SMC, do healing quest and then go straight to CoC