Depends from where you want to go to these places. All these are well connected by roads. You can drive or book a train to Kathgodam and then take a cab/bus to nainital, bhimtal and Ranikhet.
For Auli, you need to reach Joshimath from where you can book a cab or take ropeway to Auli.
Where are you traveling from? Nainital could be reached by bus from Delhi or take train from Delhi. Get down at Kadgodham (more near) or haldwani. cabs are available to Nainital. Bhimtal is nearby. can include Mukteshwar as well.
Both the places are well connected with road best way to go by bus or by car..if want to go by train then after certain point need to catch bus or cab itself .Kathgodam is the station for nanital and joshimath for auli .
From Delhi..travel towards Bareilly, then do not ho to haldwani...road is bad..go to bajpur, then kaladhungi and then nainital. from nainital, go to bhowali and hi to Ranikhet. from there you can go to auli
You can come by train to Haldwani. or enter in uttarakhand by air to dehradun airport. travel from dehradun to haldwani or Kathgodam and then take a bus and you can reach Nainital