Learn Spanish (Spain): No hay pan, así que voy al supermercado. - There is no bread, so I am going to the supermarket.

Hay is a special verb form (from haber) that means 'there is' or 'there are' to show that something exists. If you said No es pan, that would mean 'It is not bread.' To say 'There is no bread', you always use No hay.

Could I use entonces instead of así que to say 'so'?

Yes, you could use entonces (then / so) and people would understand you just fine. However, así que is a very natural connector to use when linking a direct cause (we have no bread) to a consequence (I am going to the supermarket).

Why is it written as al supermercado instead of a el supermercado?

This is a mandatory spelling and pronunciation rule in Spanish. Whenever the preposition a (to) is followed immediately by the masculine article el (the), they must smash together to form the contraction al. You will never write a el.

The translation says 'I am going', so why doesn't the Spanish use the present continuous (estoy yendo)?

In Spanish, the simple present tense (voy, meaning 'I go') is used much more often than in English to describe what you are doing right now. For verbs of motion like ir (to go), native speakers almost always use the simple present to say 'I am going' rather than the continuous form.