There is no fixed order, Considering the life of an Human being today, one cannot read and understand everything that's available. Even before starting to read, one needs to know how to read! Understanding the grammar, history, rules etc is important as it provides enough eligibility to understand. (Like the Vedangas are neccessary to understand vedas). Today, most people lack the knowledge of sanskrit and rely upon translations into some regional language. Even the veda was divided into four on the basis of Yagnya-Karma by looking at the life expectancy and time needed.
Thus, its better to start with what one likes and has the capacity to read and understand correctly. Some would prefer to read Vedas, some will start with Ramayana, Bhagvat Gita or an Agama. Some might prefer to read and learn to recite their own vedic Shakha or whatever comes through the family lineage.
In the Sarvollāsatantram first Ullhasa we find an order to understand the Shastras:
निगमादागमो जातः आगमाद् यामलो भवेत् । यामलाद् वेदसंजातः वेदाज्जातं पुराणकम् ॥ २१॥ पुराणात् स्मृतिसज्ञातं स्मृतेः शास्त्राणि यानि च । तानि गृह्माणि यत्नेन चोत्तमं हि क्रमोत्क्रमात्॥ २२ ॥
Agamas took birth from Nigamas, Yamala took birth from the Agamas, Vedas from the Yamala, from the vedas came the Puranas, Smritis took birth from the Puranas and from the Smritis we get the GrihaSutras. Take effort to understand them in the same order.
Thus the order presented is: Nigama, Agama, yamala, veda, Puranas, Smriti and GrihaSutra.
(This is from an tantric perspective, other sculptures might give different order.
Note: it's not an prescribed way to read, it's the order in which one must understand them.)