How does this whole domain transfer process work? Well, it can be complicated, but in it's simplest form, you need these three things.
1) The domain needs to be 'unlocked' at the current registrar.
2) You need the EPP Transfer Key if applicable for the TLD in question.
3) You need to read and respond to the confirmation emails that will be sent to the admin contacts.
That is that in it's simplest form. If the domain is locked (TransferProhibited) then the transfer process will fail. If the TLD requires an EPP Transfer key and you don't have it, then the transfer process will fail. If you don't read and respond to the email confirmation, then the transfer will fail.
There are also restrictions on when you transfer - the Registry may not allow you to transfer a domain too close to the initial registration date, or too close to a rewewal data, or too soon after it's been transferred in our out of a registrar - so make sure you get the transfers completed at least a few weeks before the domain is coming up for renewal. Also - the transfer process will include a year's renewal anyway, so it's best to initiate any transfer nice and early, since you can transfer at any point through the year without loosing any registration time at all.
If you do have all your ducks in a row, then the transfer process should take about 3-5 days to complete on average.