I think the OP was asking more basic questions: when, and how to do it, in particular.
On Prince difficulty and lower, you can wait to get it built after you have 2 or 3 total cities. That has the extra bonus of allowing you to research more before you build the Oracle. If you are not industrious or don't have marble hooked up, plan on chopping wood or using the slavery civic to get it done.
On higher difficulties, you will need to start Oracle earlier if you don't have marble and/or aren't Industrious, and you will therefore have to settle for a "lesser" tech from the slingshot. You will also find it hard on Monarch and higher difficulty to get both Stonehenge and the Oracle without sacrificing your empire's expansion.
As to what to pick with the Oracle ... read below. If you don't use the oracle to grab a religion and haven't already founded a religion, you should strongly consider using the Oracle's great prophet to lightbulb Theocracy and found Christianity.
Top tier Oracle techs
Code of Laws 350 beakers
Requires priesthood and writing. A good all-purpose tech that will most likely give you Confucianism and aid in your economy (and espionage points too, with BtS). The Oracle's Great Person Points will then generate a great prophet with which you can create the Kong Miao holy shrine.
Metal Casting 450 beakers
Requires pottery and iron working. Gives you access to forges and the Colossus, which is great if you are Industrious (cheap forges), have a lot of coastal cities, or just want an edge in warmongering. You can then use the great prophet to either lightbulb Theology (after picking up monotheism), crank up your forge-boosted production by settling him, or to found a holy shrine if you founded one of the early religions.
Theology 500 beakers
Requires writing and monotheism. You will almost certainly found Christianity (as it's an expensive tech for the AI to reach with traditional research) and can use the great prophet for the holy shrine. Theology also gives you a solid warmongering option, giving even non-Aggressive civs 5 xp units with a barracks.
Civil Service 800 beakers
Requires Code of Laws or Feudalism. This is hard to pull off due to the tech prerequsites. Civil Service allows you to spread irrigation to non-fresh water tiles and gives you access to Macemen once you also have Machinery, which is a long way off at this point (Machinery requires Metal Casting and is quite expensive in the early game). Personally, I think the Civil Service slingshot is overrated and hard to pull off consistently, but it is a very expensive tech you can get early. Civil Service really doesn't provide a significant short-term benefit, though, unless you are having having food production issues and need more irrigation. Also, in BtS, I have noticed that the AI civs will cancel their trades and diplomatic deals with you if you out-tech them and beat them to Civil Service.
Feudalism 700 beakers
Requires Writing and Monarchy. This opens up serfdom and vassalage, great builder and warmonger civics, respectively, and unlocks longbows. If you are a Protective civ and want to beef up your archers for offense or defense, this is a great way to go. A protective civ running vassalage (with barracks) can make a longbow with City Garrison I and Drill III (or Drill I, Combat I, and Shock) right out of the gate. This makes archery units useful offensively.
Alphabet 300 beakers
Requires Writing (although I've always thought you needed an alphabet to write ... go figure). Lets you tech trade and opens up Literature (however, in BtS, Aesthetics is required for Literature, and may be a better Oracle tech choice if early literature is your goal). Despite its relatively cheap research cost, I include this as a top tier option because it is a key component of many beeline research strategies.
Other Oracle tech options
Monarchy 300 beakers
It may seem like overkill to use the Oracle on this tech (it's fairly cheap, compared to the other techs on this list), but if you finish the Oracle too soon, or are in a higher difficulty game where you are rushing against the AI to build it, it may be worth taking. Hereditary Rule is a great civic for growing your cities by garrisoning units, which incidentally boosts your power rating and helps your relations with other civs. Monarchy also unlocks wineries, which add a ton of commerce if you have a lot of grapes in your borders.
Currency 400 beakers
Requires Mathematics. Gives your cities an extra trade route and allows you to build marketplaces. If money is tight this may be an option, but you'll make more money from building courthouses and reducing city maintenance in the early game. This is probably overkill for the Oracle.
Construction 350 beakers
Requires Masonry and Mathematics. This is a great warmonger tech for those of you needing catapults or who have access to ivory and War Elephants. It's pretty cheap to research this normally, but if you are in a war and need catapults right now, it could be worthwhile.
Calendar 350 beakers
Requires Sailing and Mathematics. Calendar centers the world map and lets you build plantations, which can be great depending on your resources. Getting Calendar early will really crank up your economy and give you trade bait for other civs. Since, in BtS, Calendar no longer obsoletes Monuments and Stonehenge, the disincentive to take it is gone. If your religious and warmonger needs are being met elsewhere, I would consider Calendar for the economic benefits.
Conclusion
Choose your Oracle tech based on your civ, location, peace/war status, and religious status. Pick a free tech that you will actually use soon.
except it's a tech that's 1½ times more expensive than Civil Service, plus it blocks off the whole Theo/DR lightbulb path so future Prophets & Artists can bulb useful stuff like Nationalism. It's also very doable on Prince level and possibly Monarch.
