After entering the library through the secret room you will find yourself in a very large room in which various items are hidden. You should start exploring the library from the right side and gradually move to the left so that you won't miss a thing.
Find the safe combination in the library
Start with the wall on your right side - there you will find Sir Reginald's armour. Then activate the gramophone on the left - it will record a music well known for the fans of previous entries in the series. Another item, Amelia's journal, is lying on the cart in the middle of the passage. Near the fireplace there is a table with Lord Croft's audiolog on it. Then walk towards the large office standing under the window ahead of the fireplace - there you will find the journal with a clue related to the safe combination. Before moving further notice that there is a chest with an important item - lighter - under the document you just picked up. It will soon prove to be needed.
Walk towards the corner of the office after picking the item. There you will find another of Lord Croft's recordings. Behind the desk, on the windowsill under the window, there is an old drawing from Lara's childhood. Now walk to the couch in the middle of the room. Behind it there is a bench with another document on it. Behind you there is a smaller desk with another item on it. Once again, don't walk away from the office since there is another document under the one you just picked up.
Now it's time to enter the first floor of the library. On the left there is a ladder. Climb it to the top and at the end of the short passage you will see a photo lying on the bookshelf. Go back down and move back to the desk under the window. Among the bookshelves you will see a movable ladder. Walk towards it and move it to the other end of the rail where you will see that the path up isn't blocked by anything. Go up and move the cart towards you in order to unlock the second gap in the handrail. Go down and move the ladder back to its original position. This will unlock access to the whole first floor of the library. Climb up and move to the end of the small corridor. On the chest there you will find another important item - a map with a key drawing. The chest is sadly closed.
Go back to the bottom of the library (you can safely jump down) and walk towards the fireplace. Use the previously found lighter to start a fire. Then use the source of warmth to unravel the ink on the map you just found. It will show you where the key was hidden.
Find the key hidden in the library
Once you check the map, the location of the key will be marked and you will be able to see it by using survival instinct. Look behind and walk towards the low drawer standing next to the bookshelf. Walk towards it from the back and move it in order to unlock the location of the master key. From now you can open chests and passages that were previously unavailable. Go back to the first floor of the library and open the chest on which the map was lying - you will find another relict which is also the second clue related to the safe combination.