If you are going via Bhubaneswar (assuming you are taking a flight) spend some time visiting the temple circuit of Bhubaneswar the day you arrive (2-3 hours for Lingaraj, Rajarani, Kedargouri temples etc) and go to Puri either via Konark on the new highway or via Pipli (a shopper's paradise) on the old route. Next day can be spent exploring chilika lake and the surrounding small villages that are known for their chenna sweets). Go to jagannath temple and check out lunch/brunch at Grand Central opposite the temple and return to Bhubaneswar via the other route (not opted for while going to Puri) on the third day.
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You can reach bhubhneshwar by train then by bus or taxi u can reach Puri . in Puri places to visit konarak sun temple Jagannath Puri temple aarti sahib gurdwara and local gurdwara Puri beach