Alors, Let's have a wee look at these examples.
"Je dois aller..." simply means that you have to go somewhere or do something i.e. it can be followed by a verb or a place. For example, "Je dois aller manger, sinon je vais être en retard."
"Je dois me rendre..." on the other hand must be followed by a place and implies that you must be in that place for a specific or official purpose. For example, "Je dois me rendre au bureau dans 20 minutes pour un rendezvous."
Now, 'un morceau' is a measure that mean 'a piece' and has no other meaning. It is a fairly general term that can be used with most nouns, i.e. you can say 'a piece of pretty much anything'.
'Un bout' on the other hand means 'a piece' as well, but it can also imply more 'the end piece' because it's second meaning is 'the end/extremity (of something)'. For example, "un bout de pain" could be "the crust/the last piece of bread/the end of the loaf". Also, in the capacity of 'a piece' it is generally only used for certain few nouns, i.e. bread, wood, land.
I hope this helps Jodi and keep up the good work!