How do I know when to write 'e' and when to write 'è'?

The distincion between 'è' (with an accent) and 'e' (without an accent) resides in the fact that the former is a verb, while the latter is a conjunction. 

'è' is thew third person singular of the verb to be

Lui è uno studente. (He is a student)

'e' is a conjunction, as in it links the parts of a sentence together

Io studio e vivo a Durham. (I study and live in Durham)

Once you spot the difference between the two, it's easy to choose one or the other depending on the context of a sentence. Just look at the sentence that you need to write: if there is already a verb, the e is likely to have a conjunction function.

Io mangio _ vado a casa

In this sentence there are already two verbs (mangio, first person singular of mangiare - to eat - ; vado, first person singual of andare - to go -)  

You will therefore write: 'Io mangio e vado a casa' (I eat and then go home)

Similarly, if you can't spot a verb in the sentence but you can spot an adjective, you will use è in its verbal form.

Il mio cane _ carino.

There is no verb in this sentence. However there is a subject (Il mio cane - my dog) and an adjective (carino - cute). You will therefore write:

Il mio cane è carino.