Officers are appealing for information about the incidents which occurred in the Kirkstall Road area at around 9.45am.
The police would particularly like to hear from anyone who may have mobile phone footage of the robbery in Harrogate or the car thefts in Leeds.
Extensive enquiries with neighbouring police forces are ongoing at this time to try and locate the suspects.
Detective Inspector Nichola Holden, of Harrogate CID, said: Understandably, the public will be concerned by the chain of events on Thursday morning, however I would like to reassure people that we are doing everything we can to locate the suspects as soon as possible.
If you have any information that can help to identify the suspects please call the police or Crimestoppers. It is important that if you have any footage or images of any of the incidents that have taken place you share it with us.
Anyone with information should contact North Yorkshire Police on 101 – select option 2 – and pass information to Harrogate CID.
If you prefer not to give your name, Crimestoppers can be contacted anonymously on 0800 555 111.