My character has a staff of fire which has 10 charges.
How quickly does my character gain its charges back?
Can I use a turn to rest and regain staff charges?
Is there something I can do to recharge it more quickly?
To answer your question: charges on weapons (presumably magical ones based on the question) are regained depending on the item. If you are using any of the items in a Hardcover book such as the Dungeon Master's Guide or Xanathar's Guide to Everything, then it should say how many charges the item has, and when those charges are regained. If the item stats do not mention charges, the item doesn't have charges. If it mentions charges and it does not say when it regains charges, assume the item does not regain charges.
That being said, for the most part, magical items regain a certain number of charges daily at dawn. The DMG does mention, however, that the DM can perhaps arbitrate a different time depending on the item (for example, a sword that is "forged on the moon" might regain its' charges at Midnight, or perhaps may only regain them once a month when the Full Moon is out). I cannot find the page number but I do recall reading this.
Moving along, as I said earlier, most items that have charges do specifically say when the charges are regained.
Now, in D&D 5e you do not take turns or actions to rest. The DMG specifies two general kinds of rest, short and long. Short Rests are 1 hour in the RAW and Long Rests are 8 hours. If the magic item is somehow dependent upon the player's resting schedule for regaining their charges, then this is what you would refer to.
Finally, magic: there is no spell, cantrip, or other magical effect (that I know of) that makes magical items regain their charges more or less quickly, excepting the open-ended version of the Wish spell which could theoretically be used to restore any number of, or all, charges to a magical weapon or other magic item that has a limited number of charges. That being said, however, your DM may choose to add or may allow you to add a spell that has such an effect, but you should discuss this with them as this would fall under house rules.
Apologies for not referencing the staff of fire specifically as I wrote this out before that edit was made.
The charges are regained at dawn
The staff regains 1d6 + 4 expended charges daily at dawn.
Additionally, there is no "Rest" action, or similar. If you wish to rest, you can take a Short Rest or Long Rest (Players Basic Rules, pg 67) outside of combat.
The staff regains 1d6 + 4 expended charges daily at dawn. (SRD V_5.1, p, 243; DMG p. 202)
Usually, that means that after an adventure day, and after a long rest, the staff will restore 1d6+4 charges. (Most wands and staves recharge in a like manner). The general design features for a "design model" standard adventure day are about 6 medium to hard encounters, with two short rests (DMG p. 84), and each adventure day will be separated from the previous day by a long rest. Short rests take up an hour, long rests eight hours, using the core rules and no optional rules.
You can't get it to recharge more quickly than that, though if your DM is only running one or two encounters per long rest, then the charges will appear to be restored more often than someone whose DM runs an adventure day closer to the design model.
As to your second part of the question, "taking a turn to rest" isn't a thing in D&D 5e, although the spell catnap (Xanathar's Guide to Everything, p. 151) provides a short rest in 10 minutes, rather than an hour. That won't make the dawn come any sooner, which is what is needed to recharge the staff.