I might be able to help. 'Ancient' Greece is obviously quite a long period of time, but I have been studying the archaic period (i.e. c. 800-479 B.C.) as part of my degree, which was a period when much Greek colonisation took place.

In short, yes, in the case of colonisation specifically, there would typically be some sort of religious element. Several of the most well-known cases of Greek colonisation (of Cyrene in Libya and Taras in Sicily, for example) attest to this. Specifically, the temple of Apollo at Delphi seems to have played an important role in archaic Greek colonisation, at least from the perspective of the Greeks themselves.

Delphi was probably the most important Panhellenic sanctuary, certainly in terms of colonisation. The Greeks viewed it as the centre of the world and went there to ask about a great many things, including anything from who a man should choose to marry, to whether to go to war with another state. The details about what exactly went on there are somewhat hazy, but there are a few things that can be pretty definitively concluded:

• Dedications to Apollo seem to have been part of the process of getting your question answered. Depending on how wealthy you were, this could include anything from simple pottery to large gold statues. Croesus absolutely inundated Delphi with gifts before consulting it about going to war with Persia (Herodotus 1.50-51, if you're interested).
• Questioners would consult a priestess-oracle known as the pythia, who would inhale fumes in order to channel Apollo and provide her answers. She is often depicted seated on a tripod and would have served for life once appointed.
• The answers given by the oracle were, at best, ambiguous. Not much can be certainly said about the way in which these answers came to the person seeking them. Some sources state that the answers were completely unintelligible to the questioner and had to be interpreted by the priests of Delphi, who accompanied the pythia. Others state that the pythia could be understood and gave answers in hexameter verse. Regardless, it is generally agreed that the answers given (the famous ones, anyway) were open to interpretation.

The extent to which visits to Delphi in cases of colonisation were a formality (in the sense that the decision to colonise had already been made before the visit) is debated. In the stories that are preserved in which someone ignores or exceeds the advice of the oracle, colonisation does not go well for them^(1). It can be said, then, that visiting Delphi was perceived as an essential part of the process of colonisation, but possibly (owing to the ambiguity of its advice) one that could be skewed by the colonisers for propagandistic purposes.

In terms of other types of 'voyage', I'm not so sure. Spartans, Xenophon tells us, had a complex system of sacrifice before setting off for war. They would sacrifice once to Zeus before leaving, only going if the sacrifice 'appears propitious'. The fire from the altar on which they sacrificed would light their way on the rest of the journey. They would also, at the borders of their land and the enemy land, sacrifice again to Zeus and to Athena, again only proceeding if the proceedings were seen to be favourable^(2).

Someone else with a bit more expertise may be able to provide more information for you, but I hope this has helped somewhat.

^(1)A lot of this information is preserved in Herodotus, whose tendency to ascribe a person's ill fate to their wanting too much for themselves may affect the reliability of these accounts.

^(2)Xenophon, Constitution of the Lacedaemonians, 13.1-13.2.

Edit: I incorrectly attributed the Const. Lac. to Pseudo-Xenophon.