Learn Polish: Często oglądam ten stary film. - I often watch this old movie.

In Polish, you usually drop subject pronouns like ja (I) because the verb's ending already tells you who is doing the action. The -m ending on oglądam clearly means 'I am watching'.

The dictionary form of the verb is oglądać. How do we get oglądam from that?

Oglądać belongs to the -m / -sz verb group. To conjugate it for 'I', you take the dictionary form, drop the final -ć, and add -m. If you wanted to say 'you watch', you would drop the -ć and add -sz, giving you oglądasz.

Since 'this old movie' is the object being watched, shouldn't ten stary film change its ending for the Accusative case?

It actually is in the Accusative case! For masculine non-living things (like a movie, phone, or table), the Accusative form is exactly the same as the basic dictionary form (the Nominative case). The endings would only change here if the masculine object was a living person or animal.

Does często (often) always have to go at the beginning of the sentence?

Not always, but adverbs of time and frequency like często usually come right before the verb. You could also put it after the verb (Oglądam często...), but putting it before sounds the most natural in Polish.