I met someone with plantar fasciitis who said certain shoes heal plantar fasciitis faster. Which kind of shoe?

Rocker shoes. The newer shape up shoes with a rocker bottom sole will help keep pressure off the area and help to expedite healing. There are also shoes with springs in the heels that seem to help.
Plantar heel pain. Basically shoes should have shock absorbing qualities, cushioned heels.
Best shoe for you. There is no one shoe that heals plantar fasciitis. The key is keeping shoes updated (changing every 9-12 months, sooner if a heavy runner) and finding shoes that are "customized" for you. Customized means shoes that fit your walking/running pattern (stability shoes, neutral shoes, etc.) as well as your foot type (high arch, low arch). Stretching & strengthening the calf muscles is also important.

Related Questions

Can plantar fasciitis heal itself if it is just a minor case of it? I got it by wearing the wrong shoes. I have normal arches but shoe caused it.

Yes. Plantar fasciitis can heal itself. Work on actively stretching the plantar fascia. Rolling a golf ball under your foot is often a good technique for stretching it. Physical therapy can be beneficial!