Most of the time, the use of mit is very similar to the English use of with. Other than that, for special phrases you will have to memorize the use of mit.

Here are a few examples: (More specialty sentences will be added as I find them)

Man hatte keinen Schluessel und musste das Tor mit Gewalt oeffnen.
Gammastrahlen bewegen sich mit Lichtgeschwindigkeit.
Bereits mit acht Jahren verfasste sie Gedichte.

In the accusative case, which asks the question where to? or wohin? and
refers to a change or movement of the object. The English equivalent
would be: into or simply in. Depending on the article of the object the possibilities are (1) for masculine article: in den (2) for feminine: in die (3) for neutral: in das or short form ins.

Of course, the German preposition ‘in’ can also be
associated with non-concrete or abstract ideas, such as a bracket of
time or a goal. The English and German equivalent would be within or innerhalb, respectively.

Here are a few common examples:

Tausende von Menschen stroemten ins Eisstadion. -[akk. movement of people]
Der Ball flog mitten in die Zuschauer. – [akk. movement of ball]
Wir waren in allen Punkten einer Meinung.
Treffen wir uns in einer halben Stunde vor dem Kino! – [dat. at a point in time]
Von Kindheit an mussten sie im elterlichen Geschaeft mithelfen. – [dat. in a trade]
Wer wohnt in dem Neubau an der Ecke? [dat.]
Herr Baur ist schon seit langem in Pension.
Lassen Sie den Prospekt ins Franzoesische uebersetzen? [akk. - a change from one language to another]