CRJ-700 flight crew reported malfunction with pressurization system during initial climb. Flight returned to departure airport.

Narrative

Shortly after takeoff out of ZZZ we got an AUTO PRESS caution accompanied by a fault light illumination on the Press Cont switch light. We observed the cabin alt rate and cabin altitude readout was mirroring our actual climb rate and altitude of the aircraft. I requested to stop our climb to our assigned 10;000' and level off at 5;000' to assess the situation and run appropriate checklists; which we did. We elected to remain on the departure while troubleshooting the issue and running the checklists. The manual pressurization procedure was unsuccessful; as the lowest we could get the cabin altitude was approximately 3;300 after multiple attempts; so we coordinated with ATC to break off the departure and requested vectors to keep us close to ZZZ while we coordinated with dispatch/company via radio to make them aware of our issue and determine the appropriate course of action. I called the flight attendants to advise them of the situation and ask if everything was normal in the cabin and the forward FA advised me that the galley service door had a very loud noise" coming from it that she didn't feel was "normal"; but everything else was ok. She affirmed that all witness marks were green; and all doors were indicating closed from the flight deck. Ultimately; the coordinated decision between myself; dispatch and operations to return to ZZZ was decided to be the safest and best action; so we coordinated with ATC; made them aware of the situation; fuel and souls onboard; etc. We did not request priority handling as there were no injuries and we had full control of the aircraft at 5;000'; and we were planning for a normal landing in ZZZ. We requested and received vectors for runway XX; as we were vectored south of the airport and completed our descent and approach checklists before notifying ATC that we were ready to be vectored onto the approach. We continued to a normal landing; and as soon as we were on the ground we received a Right Emergency Door caution message that illuminated for a minute of so before going away; then came on again and remained on until we parked at the gate. I called the FAs to ask them to inspect the door for any abnormality and was advised the door appeared closed and latched properly and that nobody was seated immediately next to it. We parked the aircraft at the gate and were met by maintenance after passengers deplaned. I debriefed with the crew and wrote up the maintenance issues for the pressurization and emergency door issues. A follow up phone conversation with the duty pilot occurred approximately 20 minutes later to discuss the specifics of the event. To my knowledge there were no reported injuries to crew or passengers.Cause: Failure of the auto pressurization system to control cabin altitude; and subsequent inability to manually control cabin pressure."