A negative with lesser contrast (from pulling) can easily be given more contrast in post production (in digital work flow after scanning or while printing in the real dark room). However, more/too much contrast from pushing is not that easy compensated (if needed/wanted).
In general pulling will give lesser grain and pushing will give more grain. Choice of film & developer might also influence this. But for now I wouldn't pay too much attention to this factor unless the results really annoy you. (BTW: grain is a dangerous subject in many discussions, often with many myths, (dis)believers fighting, false facts, emotions and many contradicting opinions ... be warned).
With colour film there might also be a shift in colour - which can be nice - sometimes also combined with cross processing. My (very limited) experience on this matter is better with Fuji films than with Kodak films.
Do experiment and do register how the film was rated, exposed, developed, printed, etc., so you can reproduce the "effect" and develop your own style. Try to change only one variable at a time, so you know what variable caused what effect.
And don't forget to have fun!!