TL;DR

SpaceX plans to launch flight 14 no earlier than September 22. It will enter a circular orbit and deploy Starlink satellites.

SpaceX should release a flight plan soon but we can expect a ~250-350 km circular orbit with a booster splashdown near the launch site. It will then deploy Starlink v3 satellites which were made specifically for Starship. Some of them were already tested briefly on the previous suborbital flight.
After some yet unknown time, but almost certainly under 24 hours, it will do a deorbit burn and reenter for a splashdown. Potential target sites are the Indian Ocean as before, the Pacific near Hawaii, the Gulf of Mexico near the launch site (overflying the US and potentially Mexico), and some strips in the northern and southern Pacific.

It is unknown whether SpaceX will try to keep this ship intact on splashdown, intentionally crash it harder, or just leave it up to chance. We don't know why flight 14 is not trying to catch the ship, there must be something SpaceX needs to work on. We can expect ship catch attempts once they are happy with everything. More reentries will help.
Personally I don't expect a return to the Gulf until they go for a catch attempt. You have all the additional risk from a reentry over populated places without the benefit of a catch attempt. The ship would be much closer to the engineers for inspection, of course.

Assuming nothing goes wrong, flight 15 could follow in under a month. The ship and booster are already assembled and being tested. We might see about one flight per month until ship reuse increases that.

Edit: Moved to no earlier than September 22.

275 km orbit for about 10 hours, with a splashdown in the southern Pacific. It will deploy 26 Starlink satellites (~45 tonnes).
The flight will reuse two heat shield tiles from the previous mission.

SpaceX has been gradually increasing the payload they carry as they get better data on ship performance. The most efficient trajectory would do an orbit insertion after half an orbit (~45 min) instead of 25 min. The steeper ascent with an earlier orbit insertion burn is likely a safety measure, staying away from orbit and controlling the reentry corridor if the orbit insertion doesn't work for whatever reason.

Edit: Some more explanation by SpaceX (this link will stop working eventually, but it should work for a few weeks)
The initial ascent trajectory matches what they have flown so far. They'll have three earlier reentry options: Directly (no orbit circularization) in the Indian ocean, after a bit under one revolution (north Pacific), and one orbit before the planned deorbit burn (south Pacific).

Still everything on track for a Monday launch.

SpaceX coverage will start here in ~2:40
NASASpaceflight coverage is running

Every orbital flight in history had at least one stage or tank that was discarded in flight - left to crash into the ocean, burn up on reentry, or remain in space. This one will be the first exception, with both booster and ship being controlled until splashdown (or at least that's the plan).

Edit: Payload deployment is underway.
Edit: Payload deployment completed.

They have contact with all 26 satellites.

Should be the heaviest object ever to reach orbit in a single mission. It's even heavier than Mir and Tiangong. Only the ISS is a heavier man-made object in space and that was assembled with many flights.

Starlink satellite looking back at Starship after deployment:

Booster landing (look for the shockwave!):

The early reentries had tons of particles falling off the vehicle - you can see them as bright sparks racing across the screen. Step by step they have reduced that rate. This reentry looked like the smoothest one so far. You just have the glow of the compressed and heated atmosphere, and that's it. This looks great for reusability.

The deployed Starlink satellites have the bandwidth equivalent of 10 Starlink launches on Falcon 9. Increasing the payload to 100 tonnes will grow that to a factor 20. That means just one Starship launch per month can easily outperform the effective deployment rate Falcon 9 reached from Florida.

Edit: Starship in front of the Sun. You can see the sound:

Starlink looking back at Starship and a couple of other Starlink satellites deployed later, engines firing, and Starship over mountains and a tropical storm:

Hot-staging from various perspectives:

Takeoff as seen from the launch tower

SpaceX is aiming at November 8 for flight 15 (only called "flight 15") and December 10 for flight 16 (called "Starlink 30-1").

Flight 15 will try to catch both booster and ship. It looks like it will not carry a payload to keep the interior of the ship as simple and predictable as possible for the first catch. If successful, it would be the first orbital flight that recovers all its hardware.

Based on the mission name, flight 16 should deploy Starlink satellites. If 15 catches the booster and/or ship, I expect 16 to try to repeat that, this time with the Starlink dispenser installed.