LEGO mindstorm - which product to use to make projects mentioned below
A basic EV3 seems to be enough for most of the basic projects, but you might find yourself in need of a handful of extra parts (gears, beams, etc). The basic kit does not include a gyroscope, but you can buy an extra one separately. The same stands for the ultrasonic distance sensor.
The alternative is the educational set which already includes the ultrasonic sensor.

All EV3 sets come with the official software, but for any serious project you should indeed look for a more capable software. For Java there is LeJOS, for C there is NXC. Both are free and open source.

The motors sold in EV3 kits are capable of continuous rotation, but also have position encoders in them, making them capable of moving with 1° precision. For more precision you can always use gearing or other mechanics to reduce movement.

Remote controlling is possible with Bluetooth (built in) or with WiFi (via an external USB dongle).