I need general informations about the Mobility/Traffic Situation in
New Delhi. How do people get from A to B and on weekends to C? The
importance of personal cars, taxis, busses, trains, etc.

Delhi, including New Delhi,has all the modes of road transport you
mention - all of them very croweded, and running on roads which by
European or US standards are in poor condition  and fairly chaotic.
(The buses are so crowded, that few people who could afford anything
better would ever take them.) In addition, there are thousands of
bicycles and tuk-tuks. The latter are 3-wheelers powered by a
motor-cycle engine, and are used like taxis. There are of course
privately owned cars, but only a small proportion of the population
can afford them.

India has an extensive rail network, which is slow by Western
standards, and again very crowded; advance reservation is often
essential - especially at holiday times - but perhaps surprisingly
there has been a computerised booking system for many years. Also, of
course, in such a large country (and with such bad roads and slow
trains) air transport is commonly used for long journeys by those who
can afford it.