Is this the brand-new Tomb Raider for the PlayStation 3 that you're talking about or a PSOne Classic you've purchased on the PlayStation Store? If you're speaking about one of the much older titles then are there objects in the level you could walk up to to activate a save? It's been too long since I've last played one of those Tomb Raider games. But, fortunately the PSOne downloads do come with an instruction booklet. All you have to do is hit the central PS button on your controller and select the option for the instruction manual. Look through it and see whether it tells you how to save your game.
After that, the game automatically saves after you snag any new collectable item or Challenge update whilst you're playing. It also performs a gamesave when you sit down at any Campsite.
You can juggle the gameslots if you wish to swap one out as you'll see the term 'Autosave X' on the save file, where X is the gamesave slot.