Can a PC worship more than one deity, and if so, what would be the pros and cons?
Yes, in theory
However, if you are looking at game mechanics of 3.5 and you are looking at a class that depends on divine magic/favour, then you would normally need to choose a patron god or "a cause or source of divine power". (PHB p. 30)
The materials for 3.5 suggest that mechanics of classes that rely on divine magic need to be prudent about who they worship - otherwise, the well that sources their magic might just run dry. Or worse!
Divine Spells - Clerics, druids, experienced paladins, and experienced rangers can cast divine spells. Unlike arcane spells, divine spells draw power from a divine source. Clerics gain spell power from deities or from divine forces. The divine force of nature powers druid and ranger spells. The divine forces of law and good power paladin spells. (PHB p. 179)
If the class relies on divine magic, the PC's alignment needs to be not more than one step away from patron god's alignment. You will find information on the Alignment in Chapter 3, Classes. For example in the case of a Cleric:
Typically, a cleric is the same alignment as his deity, though some clerics are one step away from their respective deities in alignment. [...] Some clerics devote themselves not to a god but to a cause or a source of divine power. These characters wield magic the way clerics devoted to individual gods do [...] A cleric devoted to good and law, for example, may be on friendly terms with the clerics of lawful and good deities and may extol the virtues of a good and lawful life [...] A cleric’s alignment must be within one step of his deity’s (that is, it may be one step away on either the lawful–chaotic axis or the good–evil axis, but not both). [...] A cleric may not be neutral unless his deity’s alignment is also neutral. (PHB pp. 30-32)
I don't see there being any problem with worshiping another god or gods, as long as the lore does not suggest that the gods in question are at odds with each other. This is suggested in the source materials:
The typical person has a deity whom he considers to be his patron. Still, it is only prudent to be respectful toward and even pray to other deities when the time is right. Before setting out on a journey, a follower of Pelor might leave a small sacrifice at a wayside shrine to Fharlanghn (god of roads) to improve his chances of having a safe journey. As long as one’s own deity is not at odds with the others in such an act of piety, such simple practices are common. (PHB p. 106)
To contrast the example, if a Cleric whose divine magic relies on the LG patron god Moradin LG, begins to regularly pray and worship the CE Orc god Gruumsh, they may find themselves in rather nasty pickle. This not only goes against the principle of similar alignments but would go against the lore because these two gods are in direct opposition. The outcome may be that Moradin revokes his favour, or much worse.
Gruumsh [...] harbors a special hatred for Corellon Larethian, Moradin, and their followers. (PHB p. 107)
Always look towards the alignment of the god. If the secondary god's alignment is in direct conflict with the alignment of the patron god, I would suggest that this does not work. There is a helpful table from the Hierophant section under prestige classes to determine direct opposition (DMG p. 189).
Table of Opposed Alignments
- Lawful good -- Chaotic evil
- Neutral good -- Neutral evil
- Chaotic good -- Lawful evil
- Lawful neutral -- Chaotic neutral
- Neutral* -- Lawful good, chaotic good, lawful evil, chaotic evil
- A neutral hierophant chooses one of these alignments to be the one that he opposes, for the purposes of this special ability.
- Chaotic neutral -- Lawful neutral
- Lawful evil -- Chaotic good
- Neutral evil -- Neutral good
- Chaotic evil -- Lawful good
I would suggest reading in detail the sections on Alignment and Religion in Chapter 6 of the PHB (pp. 103-109) as well as the section on Religion in Chapter 5 of the DMG (p. 143) before making any final decisions.
The final choices will be entirely up to you as a DM to decide.