what abilitys do?

[kakari]

I know that theres abilitys that raise and lower stats but
What do hardy, docile, serious, bashful, qiurky natures do

And is it possible to raise a pokemon to stats over 300 with
Those natures

I have a shiny starly with qiurky nature will it get stats over 300

[reshilegend]

These are natures, not abilities.
All of your examples are neutral natures, they don't boost or lower anything.
The future stats depend on the base stats, IVs and EVs of your Starly. Meaning you have to calculate the IVs to see if EV training would make it possible to get > 300 stats. If the IVs are high enough you could EV train your starly e.g. in speed and attack and it could get these stats.

[kakari]

Ah I wrote it wrong sorry
.. And oh okay

[reshilegend]

If you post the stats on lvl 50 here (go to the PWT/battle subway, do lvl 50 battles and check the stats) I could also calc the IVs for you and tell you whether it's possible or not.