Fallout 3/Picking up the Trail

Introduction and recap
The quest begins after the Enclave claims project purity; where you escort Dr. Li through the tunnels. (Information you may have seen in The Waters of Life is also repeated here, due to the fluid nature of the mission.)

You need to escort Dr. Li and the other scientists to the Brotherhood of Steel citadel through an escape tunnel. In the first tunnel segment, you will find a room filled with enclave solders trying to shoot you through a doorway. In the second part of the tunnel, some scientists may get killed; when going up flights of stairs, two enclave soldiers will generally have clear shots to the scientists you need to escort and will attack from behind.

After reaching the surface, Dr. Li will inform Elder Lions that she has arrived and to open the gate.

Pre-war computer access
Scribe Rothchild will give you access to pre-war computers to get information; he is located in the laboratory (and identified by the VATS compass); while he is unaware that a G.E.C.K may exist, he does direct you to the Vault-Tek computer located in the A-Ring; after checking the computer, you will find that Vault 87 has a G.E.C.K. in the non-standard equipment.

Return to Scribe Rothchild, and he will inform you of the location of Vault 87 by providing you with a map; this map also contains the locations of a few other vaults as well if you haven't seen them already.

If you want, you can proceed with some other side quests if you haven't otherwise determied the locations of the vaults:
 * Agatha's Song

Vault 87
Vault 87 is surrounded by a strong radiation field; as such you cannot safely approach the vault from the surface. However, there is an underground entrance through Lamplight caverns.

Lamplight caverns
You can enter lamplight caverns using the speech skill or using the Child at Heart perk. If you are unsuccessful, you can obtain a quest from the mayor to rescue two kidnapped children from Paradise falls.

Power Armor training
Once you enter the citadel, you can speak with Paladin Gunny to receive power armor training. During day time, he may be found in the live-fire training area.