Hey Johnny! I found this video very amusing, I struggled with a lot of the same issues on my fisrt play of Tomb Raider. The PSX version has a tutorial called "Lara's Mansion", they teach you the moves there.
You see, there's a button to "walk", and when you press it you can't fall off ledges, really useful, in order to perform running jumps (which I notice you had some issues with :p) you need to walk to the edge, press "down" once (Lara will jump back) and press "up" and the "jump" buttons at the same time and don't let go until Lara has jumped.
mled's comments