My question stems from a lack of knowledge of the empire as a whole. In Jerusalem, the Jews seemed to be able to practice Judaism and maintain their temple until it was destroyed in around 70CE. Were citizens in other parts of the empire able to decide which deity to worship?
EDIT: As a followup question, what are my choices of non-emperor worship?
The Romans were fairly okay with other faiths, allowing them to be practiced freely as long as the emperor was included and respected. In the case of Judaism, the Romans recognised that, while the emperor wouldn't be worshipped as such, he could be included in prayer. They also recognised and respected the ancient nature of Judaism, as well as recognising that banning Judaism would wreak havoc on their ability to govern Judea. The trouble with Christianity, however, was that it explicitly would not pay tribute to the emperor, causing it to be seen not as undesirable as a religion, but as a rebellious sect. Persecution of Christians stemmed from political concerns rather than religious ones.
The Romans tended to always be suspicious of monotheistic religions; the Jews /u/Quouar explained, and the eternal enemy of the Romans, the Persians, were Zoroastrian at that time, a religion with a belief in a single god.
Religion and politics, religion and war, religion and day to day life, were one and the same to the ancient Romans; they would not have made the distinction that we would today. Almost every part of every day was governed by the will of the gods, yours or your neighbours, and there were many to choose from.
Your religion would mainly be tied to your place of birth, much the same as it is today, but the beauty of the Empire was that they welcomed anyone's gods, the more, the merrier. If you were affluent, you would be exposed to many choices; if you're one of the masses, you would have to travel, meet a foreigner; or the most probable way, the army.
But the Romans welcomed all gods: honestly, whatever gods you chose, you would be free to worship them/it, as long as you sacrificed to the emperor. Christians would be singled out because they would not do so.
The short answer is as long as you paid your dues to Rome, including sacrifice to a certain god or emperor, your own religion mattered little to the Empire, just behave.
Yes and no. On one hand, worship of new gods and the keeping of new cults was widely accepted throughout Rome, with the caveat that they didn't disrupt wider society. A key issue in this regard was the obligation of Romans and Roman-dominion residents to sacrifice or otherwise revere the 'imperial' gods; Jews were particularly troublesome concerning statutory worship, such as of the Imperial Cult (either literally worshipping a god-Emperor, like Domitian, or the 'genius' ('the spirit') of the emperor, such as with Augustus (before his posthumous deification).
Jews could get around this by sacrificing to their God 'on behalf' of the Emperor and Rome, but this was still a problematic affair, especially with early Christians. The problems between 'religions' (strict 'religions x and y' are not really applicable in Roman society) seem to revolve around what you don't do, more than anything else, although this question is further complicated by other social concerns (for example, if your 'religion' was seen as especially perverse/weird, you might find day-to-day life a bit difficult).
Sources and further reading:
J.K. Hardin (2008), 'Galatians and the Imperial Cult: A Critical Analysis of the First-century Social Context of Paul's Letter'. See esp. page 104.
You may also find /u/mp96 's post in this thread interesting/useful also.
I have a follow up question. I've been told that during that particular time period and place, religion was more part of your ethnicity than it was the personal choice/journey it is today, and so people didn't really convert to other religions without also experiencing a major cultural shift. Is there any truth to this or did I misunderstand something?
Yes and no - Rome was always incorporating other religions and gods. As has been stated, as long as the emperor was included, the Romans were fairly tolerant. Judeo-Christian persecution in the empire stemmed, in all honesty, from ignorance. Christians refused to pay respect to the emperor (I highly suggest you read Pliny the Younger's letter to Trajan if you have interest in this subject) out of belief. But the Romans looked at Christianity as a rebellious sect of Judaism. The "INRI" you see on most crucifixes stands for "Iesus Nazarenus rex Iudaeorum", which is Latin for "Jesus of Nazareth, King of the Jews". So, you would have a little bit tougher of a time getting along as a Judeo-Christian if you were devout enough to refuse to include the emperor, but as a pagan, it wouldn't be too difficult (as long as you gave the emperor his "dues", so to speak). In Apuleius' "The Golden Ass", Lucius joins the Cult of Isis at the end of the story, an Eastern cult centered around the Egyptian goddess which was quite popular in Roman culture. Also, if you've ever been to Pompeii or Ostia, you can see a Mithraeum, where followers of the Cult of Mithras carried out their... interesting rituals.
In short, Romans were accepting of other religions, to a degree. In terms of polytheism, they were often incorporating other gods, goddesses and cults into the Roman pagan religion. A very devout monotheist would have a tougher time, though it wouldn't be impossible.