Currently it seems that "Service Request" is the default Ticket Type, and there doesn't appear to be a way of setting a Ticket Type by Request Type (or any other way). Nor does there seem to be a way of changing the default from "Service Request" to "Incident".
Am I incorrect, or is that the way it is?
An Action Rule is the only way to achieve this, simply create a new action rule within Setup > Processes > Action Rules, click New and follow these screenshots:
You can change the rule trigger to apply when a ticket is updated but this will override any ticket type change a technician may make so it's best to stick this to only at ticket creation.
Note that if you're using the Criteria match-up as the Ticket Type like the above, this will apply to all tickets that the rule triggers on. If you want this to be specific Request Types or users or anything else, then change this according to your requirements.
Use the Modify Ticket and click the Edit button, then change the Ticket Type to the one you want it to change to. Don't forget the 'Change to' selection or it won't do anything!
If you need anything else changing in the ticket then add the change here too.
Once done, click save and do some tests with the criteria you've set.
Congratulations, you are now an Action Rule Entrepreneur! 
