Simplest shims to use would be small washers. (I’ve done this and it works for me.)
Agreed. For the price of cheap plane with a blade, I’d suggest a Surform plane. Handy to have around without the hassle of keeping a blade razor sharp.
For a lightweight interior door, I have just used bits of card stock trimmed to size. It gives an even shim thickness and there is no fuss in putting the holes in it.