Originally, it seems that the ideal was to learn Tanach straight, probably with Targum.
Eventually, Jewish Ed [at least in Ashkenaz] pushed Humash and Rashi in lieu of covering all Tanach. There are many trade-offs of course. To go wide on Tanach, vs. to know Humash with a sampling of Midrashim hand-picked by Rashi himself from Talmud and Midrash?
Here's where I'd start, with that portion of Tanach in the Liturgical cycle
D. Selections from T'hillim - especially those that appear in the siddur and Mach'zor
This covers nearly 50% of Tanach using its most familiar passages.
But, which commentary? There are so many to choose from, here are a few suggestions
A. Rashi [of course]
C. Living Torah [R Aryeh Kaplan
D. Soncino Humash
E. Hertz Humash
Soncino Nach is really Helpful for B'qiut in Nach. Also the M'tzudos are useful.
Philip Birnbaum has a nice edition of the 5 M'gillot. Artscroll also has a nice version of M'gillot as well as several editions of T'hillim.
For more B'qiut in Talmud, I would recommend Torah T'mimah on Humash and M'gillot.
I've heard that Hakk'sav v'Hakkabalah is also very good. Malbim adds the Midrashei Tannaim to the mix.
Shalom and Regards,