Learn Turkish: Yeni bir tişört alıyorum. - I am buying a new t-shirt.

In Turkish, when you use an adjective with 'a' or 'an' (bir), the adjective comes first! So we say yeni bir tişört (literally 'new a t-shirt') rather than the English order 'a new t-shirt'.

Doesn't the verb almak mean 'to take'? Why is it translated as 'buying' here?

You are right, almak primarily means 'to take' or 'to get'. However, in everyday Turkish, it is also the most common verb used for 'to buy'. It is similar to saying 'I am getting a t-shirt' in English.

Why doesn't tişört have an object suffix on the end since it is the thing being bought?

We only use the specific direct object suffix (the accusative case, which would make it tişörtü) for specific things, like 'the t-shirt'. Because this is just a general, non-specific object ('a new t-shirt'), it stays in its plain dictionary form.

How do the suffixes in alıyorum work?

It starts with the verb root al-. Then we add the present continuous tense suffix -ıyor (notice how it uses ı to match the a in the root according to vowel harmony rules), and finally the 'I' personal suffix -um.