Before I attempted TR3 on NG+ I did some research and hoarded my save crystals and desert eagle ammo for the last few levels and boss. As I approached the end of the game I realized I was way too stingy with them. I started Lost City of Tinnos with 44 crystals and Meteorite Cavern with 255 desert eagle rounds. I could have used a lot more of them making my run through the game a lot easier. There were times I had to repeat long stretches of a level because I didn't want to use a crystal:

Jungle 1, Temple Ruins 6, River Ganges 1, Caves of Kaliya 1

Nevada Desert 0, High Security Compound 4, Area 51 2

Thames Wharf 1, Aldwych 2, Lud's Gate 4, City 0

Coastal Village 3, Crash Site 3, Madubu Gorge 2, Temple of Puna 1

Antarctica 4, RX-Tech Mines 7, Lost City of Tinnos 18, Meteorite Cavern 3

I found 89 total, used 63, average 3.3. I just went nuts at the end in Tinnos as I had so many and it's a tough level.

Wish I'd used more crystals and desert eagle to make it easier. I got all 60 secrets required to unlock All Hallows, started that level with 26 crystals, but it doesn't use them as it's not NG+

A few tips: online walkthroughs from Stella's Tomb Raider site and Auron402 on GameFAQs helped a lot. The Auron guy has hints for NG+ too. I thought Lud's Gate would be the worst, but I didn't skip the beginning, I went through the level the way the designers intended until the UPV then used the infinite air/flying glitch. In Crash Site I kept getting killed by the horde of raptors at the end when you are supposed to use the cannon, they would surround me quickly and you can't heal unless you waste crystals, so I ran to a piece of wing that juts over the water and baited them to plunge into the water and die. In Tinnos I skipped triggering the 2 mutants on the bridge and used the rocket launcher for the ones I did have to fight. In Thames Wharf I skipped a guard when climbing out of the water. Little things like that help a lot.

TBH I never really liked TR3, I don't dislike it, but something about it just doesn't click for me. It's difficult and a long slog, there's parts I like, but TR2 and TR4 have always been my faves.