Mineral oil will never dry1, so you need to strip it from the surface of the wood with a solvent before you can safely overcoat with a 'drying' finish — one that dries and cures, which includes polyurethane.
A non-drying oil within the surface wood fibres can prevent the formation of a good bond between the finish and the wood, leading to poor durability and even peeling. And it can interfere with the proper drying of the finish too, leading to it being permanently sticky or gummy — the same effect is seen with some tropical woods which have a natural oily/resinous compounds in them, and just like with those the solution is to clean the surface of the wood.
You would do this using solvent2 and rags or paper towels, and plenty of them (so that you're physically removing the dissolved oil, not just spreading it around after it has been picked up by the cloth/paper). Just be thorough and work methodically over the surface and it shouldn't be too tough a job although fair warning, cleaning like this does use a surprising amount of solvent.
Would there be any better ways to finish it now that I have put mineral oil on it? Shellac? A drying oil?
You could in theory at least use shellac here, as many modern users of shellac who still pad it on have switched to using mineral oil from the traditional linseed oil as a lubricant on their application pads. But for safety it would still be best to to clean the oil from the surface.
A drying oil applied over wood already wet with a non-drying oil is a recipe for the drying oil to never dry.
Not related to your main query:
I sanded it up to 600 grit.
There's rarely any benefit to sanding beyond about 320 grit if the wood will be finished conventionally, and with a film finish you can get away with sanding only to around 150 occasionally, although 180-240 is the more usual range.
600 grit is fine enough to be considered a polishing grit, and once a film finish goes on the wood the finish itself becomes the final surface and provides the polish, so there's no need to polish the wood beforehand.
1 Mineral oil on wood appears to dry because it wears from the surface with use and handling, and may sink in a little deeper (very little, on the order of a few cells in the wood).
2 You can use mineral spirits, paint thinner, VM&P naphtha, xylene or toluene (which are all part of the same spectrum of hydrocarbon solvents) or acetone. Note that if using MS the low-odour or "odourless" kind is less effective.
