What kind of paper should one use to print custom cards?
at least 110# card, probably not above 200# card. 110# is thicker than my playing cards, as an FYI. I've used it for quite shuffleable cards for playtest games. 

Once you get past 200# stock, you're into stuff too stiff for comfortable shuffling.

A high rag content is good, as well, as it's more resilient than pure wood-pulp stocks. High clay content is bad, for the same reason, and because it results in longer drying times, higher chance of backs being marked, and in my experience, higher chance of laser printing wearing off.

Note that if you're doing print and play for long term use, it's better to use some form of ink printing rather than laser/photocopier, due to face wear. (Remembering that inks soak in, while laser toner is actually a surface deposit, and thus wears off.) 

Remember: for small runs, odds are that the copy shop will simply photocopy (usually using a laser copier) the material. If you can afford offset for the run, it's probably the most durable.

For playable home use, 110# white cotton bond no-clay card is fine, especially if using an inkjet.