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There exists hardly any reading sequences of the puranas. One can start with the one he likes. It is because all the eighteen major puranas briefly or broadly deal certain similar contents (creation, dissolution, lineage, etc.) and describe past events taking into account a central deity of the Trimurti. So a single purana can be found narrating tales of both Rama and Krishna and events of Satya, Treatya and Dwapara yuga along with certain nature of the age of Kali. Some also describe future events. But all the puranas are considered to be the creations of Vedavyasa. Hence, their literal creation times are probably same and limited to the Dwapara Yuga.

It is the literature of itihasa(history) that are associated with different time periods. For example, Ramayana is associated with Tretaya yuga and Mahabharata is of the Dwapara Yuga. But if you need a particular sequential order of the puranas from the oldest to newest, then it would be something like below as described in the Vishnu Purana: 

The Order:

1. Brahma Purana

2. Padma Purana

3. Vishnu Purana

4. Shiva Purana

5. Bhagavata Purana

6. Narada Purana

7. Markandeya Purana

8. Agni Purana

9. Bhavisya Purana

10. Brahmavaivarta Purana

11. Linga Purana

12. Varaha Purana

13. Skanda Purana

14. Vamana Purana

15. Kurma Purana

16. Mastya Purana

17. Garuda Purana

18. Brahmanda Purana