How does a rocket know when to do apogee kick motor. Do they orbit once to find highest point?
There are multiple methods by which a satellite may determine where it is.
The traditional approach is ground tracking. As David Hammen has mentioned, in this case ground stations detect where the satellite is and how it is moving. This information is then used to calculate the orbit. If necessary, the satellite may be told about this orbit, but more likely the information is used by the ground station to decide what commands to send to the satellite and when. A downside is that this information is only available when the satellite is within the field of view of a ground station, and it is rather imprecise (several km for Low Earth Orbits and 50 km for geosynchronous orbits). For Low Earth Orbit the Global Navigational Satellite System (GNSS) is more accurate.
The NASA Tracking and Data Relay Satellite System (TDRSS) adds to this a constellation of satellites in such a high orbit that with just two satellites one can provide tracking data coverage for 85% to 100% of low earth orbits (Larson and Wertz), It is also much more accurate with accuracy of about 50m (3 sigma), and commonly used.
There are several autonomous or semi-autonomous systems, where the satellite determines its position without relying on a ground station. Some are in common use, others are experimental.
The Global Navigational Satellite System (GNSS), such as GPS, GLONASS, GALILEO, works well in Low Earth Orbit (nowadays used a lot), difficult in Geostationary orbit, unavailable in extraterrestrial orbits.
The Microcosm Autonomous Navigation System (MANS) is fully autonomous. It uses attitude-sensing hardware, looking at Sun, Earth, and Moon to provide real-time attitude and position information. The principle is similar to determining your position by looking at several landmarks. This method has been tested, but is not in common use. It could be used for spacecraft in solar orbits or orbits around other planets as well. For a novel method based on an x-ray sextant-like technology, see Is NICER/SEXTANT the first civilian "spacecraft" to determine it's own position in space without GPS or uplinked data?.
In principle, a satellite in Earth orbit could use landmark tracking to determine its position, but this is not in common use for satellite orbit determination (it is, however, frequently used for correcting the geolocation of images taken from satellites). Determining the distance to the transmitting ground station by signal delay is easier (I don't know if this is used in practice), but has the same limitation as traditional ground tracking in that it only works when the station is in view.
Source and further reading: Larson and Wertz, Space Mission Analysis and Design, Third Edition (2006), §11.7.2 (page 501–507).
Most spacecraft do not know where they are. These spacecraft used time-tagged commands to initiate a burn, with the command either generated pre-launch or transmitted to the spacecraft from the ground. This approach does have issues, as witnessed by several spacecraft that have performed the right command at the wrong time.
Some spacecraft do know where they are, using a combination of data from global navigation satellite systems such as GPS and from inertial navigation sensors (INS). Relying only on INS is problematic; it's called ded reckoning (apparently short for deduced reckoning), but also sometimes called dead reckoning. As in using ded reckoning is a sure way to ensure that the spacecraft will soon be dead.
There are problems with GPS-only solutions as well. The observations aren't as frequent, there are issues with reflections ("multipath"), and velocities, which are critical for orbit determination, are not very accurate. Using a hybrid solution requires a rather complex Kalman filter, making the flight software expensive to develop, verify, and validate.
The complexity of the software and the need for expensive sensors is what makes the non-navigated spacecraft solution rather appealing.
They don't know - they are injected into their path with a certain energy that determines their final trajectory at a certain height. Extremes are falling back to earth or having hyperbolic trajectories i.e. escaping earth's gravitational field. Furthermore, the inclination of the orbit can be used to control their relative placement re. to the earth's rotation for example resulting in so-called sun-synchronous orbits...