Aku/hiraku are intransitive verbs for "to open" using the same kanji. The transitive counterparts are akeru/hirakeru.
I. AKU/AKERU (開く & 開ける)
I think of aku/akeru (開く & 開ける) as referring to "making an opening" so there's a resulting hole or space. They are also used metaphorically for time expressions. So we can use it for doors, windows, bottles, cans, bags, boxes, eyes, mouths, time, etc.:
"mise ga aite-iru" = the store is open
"mise ga 9-ji ni akeru" = the store opens at 9:00
"mise o akete-iru" = (someone) is opening the store
"mado o aketara," = if you open the window,...
"kabe ni ana o aketa" = he made a hole in the wall
"doa ga aite-iru" = the door is open/ajar
"aita kuchi" = "open-mouthed" "agog"
"bin ga akanai" = I can't open the bottle
"me ga aite-inai" = his eyes are not open
"chakku ga aite-iru" = your fly is open
II. AKU/AKERU (空く & 空ける)
You can also use "aku/akeru" to mean "open" in the sense of vacant/free, but it's a different kanji, namely 空く & 空ける:
"kono seki ga aite-imasu ka?" = is this seat free?
"aite-inai heya" = an occupied room
"heya ga kanzen aite-imasen" = fully booked (hotel)
"ashita aite-imasu ka?" = are you free tomorrow?
III. HIRAKU/HIRAKERU (開く & 開ける)
In contrast, hiraku/hirakeru (開く & 開ける) is more like "to open up" so it's used for things with moving parts like umbrellas, and "to open out/wide/apart" in the sense of spreading things so for books, flowers, hands, (erm) legs, gates, shutters, collars. It's also used metaphorically for opening up your mind/heart/eyes. I've also noticed it being used for opening up computer files.
"mise o hiraku" = to open up (start) a business (cf. aku/akeru = to open for business)
"sensu o hiraku" = to open/unfold a fan
"kasa o hiraku" = to open an umbrella
"tsutsumi o hiraku to" = when you open the wrapping/parcel
"kyoukasho no 3 peeji o hiraite kudasai" = pls open to page 3 in your textbook
"mon wa hirakanai" = the gate won't open
"ashi o hiraite kudasai" = please spread your legs
"kaigi/tenrankai o hiraku" = to hold a meeting/exhibition"
"yatto kokoro o hiraite-imasu" = she's finally opening up to us
"shiten o hiraku tsumori desu" = they plan on opening up a branch office
"fairu o hiraku" = to open up a file
"sakura issei ni ga haita" = the cherries bloomed all at once
Did I miss/misstate anything?