"Which day is Uposatha?" is a good question and very importand for Bhikkhus who even need to ask if not knowing. Since it is not easy to answer, best keep the Uposatha always, like Upasaka Lanka already mentioned in his answer.
Actually, althought the Buddha allowed his monks to learn the calculation of the full moon and half moon days, there seems to be no (public/broadly know) transmission.
Since the day depends on the certain location (note that the day starts at sunrise not at 00:00! to understand, which can be very different), althought the event of full or empty is equal for the whole world, is best to request or inform you from the Bhikkhu-Sangha where you took/asked/declared refuge and received the precepts or you follow, if not avaliable, your near personal community.
The simple counting, starting from the full moon day, is eight and then fourteen/fifteen days. So it matches 29 Days a month. Since a moon-month has a little bit more than 29 days, different traditions compensate it in different ways.
As for the traditional observing and gathering days (note that the more often or even always, the better: To the Sakyans (on the Uposatha)) they are held "weekly" (see Upasaka Suminda Sirinath S. Dharmasena answer) at large but also on further days:
pakati-uposatha
pati-jagara-uposatha
pati-harika-pakkha-uposatha
The times for keeping the pakati-uposatha are the 5th, 8th, 14th and 15th of the waxing moon and the 5th, 8th, 14th and 15th of the waning moon.
The times for keeping the pati-jagara-uposatha are the five days of the waxing moon, i.e., the 4th, 6th, 7th, 9th, and the 13th, and the six days of the waning moon: the 1st, 4th, 6th, 7th, 9th, and 12th or 13th. That makes eleven days in a month for the observance of this type of Uposatha.
The four months of the rainy season, or vasso, starting on the first night of the waning moon of the seventh month and ending in the middle of the eleventh month, is the period for observing the pati-harika-pakkha-uposatha.
The commentary to the Raja Sutta explains that the pati-harika-pakkha-uposatha is the Uposatha that is observed continuously throughout the three months of the rains. If one cannot keep the observance for the full three months of the rains, then it should be kept for one month, from the first day of the waning moon of the tenth month to the eleventh month. If one is not able to keep the observance for one full month, then it should be kept for a half month, from the first day of the waning moon of the tenth month to the end of that month. Any period of this observance is called pati-harika-pakkha-uposatha.
Some texts define (the time for keeping the pati-harika-pakkha-uposatha ) as the five months from the seventh month up to the eleventh month.
Some Acariyas say the three months are the seventh, eleventh and third months.
Yet other sources explain that the four days, i.e., the 7th, 9th, 13th and 14th, both waxing and waning, are the only days for the observance of the pati-harika-pakkha-uposatha.
(source)
In regard of which precepts are observes, the questions list misses abstaining beautifications and participation on entertainments. Upasaka Lanka had listed them all well in his answer. But next to the many commentaries and explainings deriving from monks Vinaya, the Sutta in its regard might be the best: Uposatha Sutta: The Uposatha Observance.
Generally possible useful help on how might be found here: How to come to be able to observe precepts?
There might be found answers to other details when searching for Uposatha here. How ever, best to speak direct and gain advices from those successfull practice them (of which are maybe less and not always to find).
In regard of the detail question "does this need to be vegetarian to fit in with the first precept?": Not unconditionally. Importand not to kill for yourself, bring others by words or signs to kill and not to feel joy or acceptance, mental approve, for killing. The last includes not to eat meat if a living being was killed for you personally (if seen, heard or suspected). Eating what ever is not alive, does not carry any unskillfulness within it. Killing, urge to or agree, does and whould for the bodily and verbal action cause a break of the first precept, and bad effects (kamma) in all three ways of actions.
May you put effort on the foundation for liberation and observe the Uposatha eager, and if not done or forgotten, todays event, now, is for sure the best time to start or to re-start.
Sīlena sugatiṃ yanti.
Sīlena bhoga-sampadā.
Sīlena nibbutiṃ yanti.
Tasmā sīlaṃ visodhaye.
Through virtue they gain a good stream.
Through virtue they gain wealth.
Through virtue they come to liberation.
Therefore we should clean our virtue.
Anumodana
[Note: This is a gift of Dhamma and not meant for commercial purpose or other low wordily gains by means of trade and exchange.]
