This page explain the differents gameplay of challenges rooms, if you want to keep the surprise, don't read this. 
Oh, and don't forget, you're free to choose which part you want to start. :)
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---------------------------------------------------[Spoiler]------------------------------------------------------

Tomb Raider I :

Lost Valley : Shooter

Colliseum + Thor Room : Puzzle + Shooter

The Citern : Platform / Agility

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Tomb Raider II :

Venice : Platform / Search

Maria Doria : Search / Agility

Thibetans Foothills : Shooter + Platform + Strategy

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Tomb Raider III :

Jungle : Platform / Agility

Nevada : Search

South Pacific : Puzzle + Shooter

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Tomb Raider IV :

One Adventure : Agility + Puzzle + Search + Platform + Shooter

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Tomb Raider V :

Rome : Search / Platform

Base : Search / Platform

Gallows Trees : Search

13th Floor : Agility + Shooter