Maximum (positive) rapport is +5500 rapport points. Minimum (negative) rapport is -5000 points. There are 8 rapport levels: (0) "Disdainful"; (1) "Irritated"; (2) "Wary"; (3) "Cordial"; (4) "Friendly" (about +1000 Rapport); (5) "Close"(about +2000 Rapport); (6) "Allied" (about +3000 Rapport); and (7) "Companion" (about +4000 Rapport).

I have unlocked the two companions (Mirri and Bastian) on 10 characters so far. Been trying to understand what raises Rapport for these companions. Been mostly working with Mirri. Here's are things that raise Rapport for Mirri, with numerical values. Obviously, Mirri must be unlocked and deployed when you do these things. My 10 characters who unlocked Mirri got her to maximum level (level 11) on Fighters Guild, Mages Guild, and Undaunted very quickly. "Rapport" is character-specific, not account-wide.

- Handing in a Fighters Guild daily quest . . . 125 Rapport points [Bastian gets 125 points per Mages Guild "daily" quest]
- Handing in Ashland Relics daily quest from Numani-Rasi in Ald'Ruhn in Vvardenfell . . . 125 Rapport points
- Entering a daedric delve or public dungeon . . . 10 Rapport points
- Entering Clockwork City (once a day) . . . 10 Rapport points [Bastion gets 10 points once per day for visiting Eyevea or Atreum]
- Looting a psijic portal . . . 5 Rapport points
- Excavating an Antiquity . . . 5 Rapport points [Bastian gets 5 Rapport points for Scrying]
- Visiting Collections* . . . 5 Rapport points (on first visit, Mirri makes comments, but you don't get points; on subsequent visits, get points)
- Reading a book . . . 1 Rapport point
- Crafting an alcoholic drink . . . 1 Rapport point
- Killing a goblin . . . 1 Rapport point
- Killing a snake . . . 1 Rapport point

*Collections for 5 points confirmed: (1) Wrothgar Relics; (2) Northern Elsweyr mural at House of Histories; (3) Western Skyrim music collection at Bards College; (4) Southern Elsweyr Tapestry at Sanchal Palace; (5) Artreum College of Psijics. I assume other collections will also give 5 points, but I did not confirm.

Most of these have a cooldown. Seems to be 5 minutes. In other words, when crafting an alcoholic drink, you'll get 1 point for the first drink, then you won't be able to get another point (from drink-crafting) for at least 5 minutes. If you Scry, you won't get another 5 rapport point for Scrying unless 5 minutes elapses.

- Completing some companion Achievements advances Rapport by the point value of the Achievement.

- Quests given to you by your companion give you the most Rapport points (500 points each):
* First companion quest (received when Rapport becomes "Friendly," about 1000 Rapport points): 500 Rapport point upon completion
* Second companion quest (received when Rapport becomes "Close," about 2000 Rapport points): 500 Rapport points upon completion

After completing the second companion-quest, you get Mirri [or Bastian] as a "house guest." Placing the companion as a House Guest in your house yields the "Room for Two" Achievement. (But no Rapport, since companions are not active in houses, I guess.)

I spent a few hours "grinding" Rapport on the test server (PTS). I concluded that it's not worth the time and effort. I have decided to just let Rapport develop slowly (without any grinding) as Mirri and I go adventuring around Tamriel. You might try doing the Fighters guild daily (+125 Rapport points) and the Ashland Relic daily (+125 Rapport points) every day.

The Rapport indicator is scaled from a minimum (negative rapport) of -5000 to a maximum (positive rapport) of +5500. So the full scale of the Rapport indicator is 10500.

When going for an Antiquity, deploy Bastian when you Scry (for 5 Rapport points to Bastian), then swap to Mirri when you Excavate (for 5 Rapport points to Mirri).