1. Outskirts means the outer parts of an area. It is normally used to refer to the outlying areas of a city, but it could have other uses as well. Outskirts is the normal usage. I checked and there is some mixed information about whether outskirt can be used. I would stick to outskirts. Note that the "s" at the end of outskirts does not refer to a plural form, it is just a noun representing a particular location. For example - "We lived in the outskirts of Sydney", meaning we lived on the outlying or outer border area of Sydney.
2. Suburbs refer to the residential areas outside a city centre. There is a slight difference to outskirts. In point 1 above, it means on the outer edges, whereas in point 2 it means not in the centre, hence a subtle distinction. For example, "We lived in the suburbs of Sydney", means we lived outside the central city area. It may not necessary mean that we lived on the outer edge of the city as in the first case. This, as well as the fact that outskirts can be used in other circumstances, for example, you could say "The explosion was heard even at the outskirts of the firecracker factory", meaning at the outer edge of the factory area.
The above is my understanding, I'm not sure if the two terms are interpreted slightly differently in the US.