The weekend plan
Day one
Start on the walls themselves — the 2½-mile circuit is the single best orientation to the city, and the stretch from Bootham Bar to Monk Bar gives a clear view down onto the Minster's towers. Drop down into the centre for the Shambles and the tangle of narrow "snickelway" alleys around it, then take the Minster itself in the early afternoon, when the tower climb (275 steps) has usually thinned out. Evening in one of the pubs backing onto the Minster precinct, several of which claim histories going back centuries.
Day two
Pick one museum rather than both: Jorvik Viking Centre reconstructs the Viking-age city excavated directly beneath modern York, while the National Railway Museum, a two-minute walk from the station, holds one of the largest train collections in the world and is free to enter. Either way, save an hour for the riverside path along the Ouse and a look at Clifford's Tower, the surviving keep of York Castle, before heading home.
What to see
Beyond the two-day plan above, a few things are worth building time around rather than rushing past. York Minster is one of the largest Gothic cathedrals in Northern Europe, and its stained glass — including the Great East Window, the largest medieval stained-glass window in the world — rewards slowing down inside, not just admiring the exterior. The city walls are the most complete medieval defensive circuit in England, and walking the full loop (rather than just the photogenic Bootham-to-Monk stretch) takes under two hours at an easy pace.
A slower alternative: the snickelways
If the Shambles feels too busy, York's snickelways — a network of narrow medieval passages threading between the main streets — are the same architecture with a fraction of the foot traffic. There's no single route to follow; the point is getting slightly lost between Stonegate and the Minster.
How to pace it
Don't try to fit Jorvik, the Railway Museum, the full wall circuit and a Minster tower climb into one day — each one is a genuine hour-plus commitment, and stacking all four turns a walkable, unhurried city into the exact rushed checklist York is well-suited to avoid. Two of the four, plus real time in the Shambles and on the walls, makes a better weekend than all four done at a jog.