Conventional automatic transmissions have a torque converter which uses a viscous rather than a rigid coupling to connect input and output. This performs a similar function to a friction clutch but doesn't require manual engagement. Also automatic gearboxes are based on epicyclic gear trains so all the gears are permanently meshed with each other and ratios are selected by locking different combinations of the sun, planet and annular gears so there is no need to match different shaft speeds to each other.
Sequential gearboxes use a combination of straight cut (as opposed to helical) gears and dog clutches with gears engaging by sliding along the input and/or output shaft so the gears never go fully out of mesh with each other but rather are either locked to or freely spinning on their respective shaft. Again this means that it is not necessary to match input and output speeds, however a clutch is still required to start from a standstill. Motorcycles, for example typically have sequential gearboxes and hand operated clutches.
The more modern type of auto/semi-auto gearbox typically uses twin wet plate clutches to change between pre-selected gears in this case the gear selection and clutch engagement is done by actuators which are controlled by a computer. In many cases they can be operated as fully automatic or semi automatics where the driver shifts up or down by operating a paddle or lever. So while they do have a clutch is it not manually operated to change gear. Although, as in a sequential gearbox, manual clutch operation may still be required to start from a standstill often this will be an additional paddle or paddles. Although many semi-auto transmission will also incorporate an anti-stall system which automatically releases the clutch if the revs drop below a certain level at low speed.
So from the perspective of the driver there are 3 main cases.
Full auto (either conventional auto or modern semi auto in full auto mode) : press accelerator pedal to go and the brake to stop and the clutch (or equivalent) sorts itself out automatically eg with automatic transmission if you brake to a stop the engine won't stall so no manual clutch operation required.
Sequential/semi auto : manual clutch operation is required to start/stop but not to change gear. Often this will be a hand clutch.
Manual : manual (usually foot pedal) clutch required to start and to change gear.
These cover the most common systems, although there are various other transmission types around.
Sequential gearboxes are most common on motorcycles and some classes of competition cars (especially those heavily adapted from road cars (eg rallying).
Most conventional road cars have either conventional automatic or manual transmissions.
Computer controlled semi-auto/auto transmissions are mostly found on high-end sports and racing cars eg F1 and GT formulae.
There are also automatic transmissions which allow for an approximation of manual gear selection eg the Porche Tiptronic system and it is also possible to automate a conventional manual transmission, although this is a lot of complexity for no real gain.
For the sake of completeness there are also centrifugal clutches which only engage above a certain engine rpm but these are only really found on things like chainsaws and self propelled mowers. Also steam piston engines don't require a clutch as they can develop torque from zero rpm.
It is also worth noting that for a driving simulation, unless you have really good haptic feedback in the control hardware it will be very difficult to simulate the feel of a manual clutch.