> Do you think this issue would make interstellar travel all but impossible?

No, I don't believe so. The major hurdle to be solved is developing much, much more efficient engines. Radiation shielding has pretty good solutions already.

> Is shielding from radiation typically lead in composition (like the lead aprons they give you for medical X-rays), or are there other kinds or technologies?

The best material depends on the kind of radiation. For X-rays and the like (relatively low energy), lead works great. For galactic cosmic rays containing lots of proton and neutrons, things with a high hydrogen content (like polyethylene plastic) can work better and weigh less.

Water is also an excellent choice: It's decent at radiation protection and the ship will need a lot of it anyway (for drinking, cooling, growing food, possibly for the engines). Lining the outer hull with a layer of water could work. Or the ship could have a radiation vault encased in a thicker layer of water for the crew to take shelter in.

> I would imagine that lifting lead into space for this purpose would be prohibitively expensive.

The whole ship would be much, much heavier than the lining alone. If it is too expensive to launch the ship, dropping a bit of shielding mass won't make much of a difference. Without a breakthrough in the launch cost to Earth orbit, even regular interplanetary travel remains a pipe dream. SpaceX's Starship program aims to change that in the near future.