Just wondering How do Hindus decide which gods or goddesses to worship?
As per chapter 22 (The Lunar Dynasty) of Kurma Purana, Kartavirya Arjuna's sons also had similar question and they approached Saptarshis to get an answer.
This is what Vashishtha and other sages told them:
One worships the god that one chooses. There are no rules in this regard. All gods yield the desired fruit, if properly worshiped.
Further Vashishtha and other sages said:
To the extent that there are rules, there are as follows:
Kings worship Vishnu, Ishvara (Shiva) and Purandra (Indra);
Brahmanas worship Agni, Aditya (Surya), Brahma and Pinakdhari (Shiva);
Devas worship Vishnu;
Danavas worship Trishuldhari (Shiva);
Yakshas & Gandharvas worship Soma (Chandra, the Moon God);
Vidhyadharas worship Vagdevi (Saraswati);
Sidhdhas worship Hari (Vishnu);
Rakshasas worship Rudra Shankra (Shiva);
Kinnaras worship Parvati;
Rishis worship Brahma and Trishuldhari Mahadeva (Shiva);
Women worship Devi, Vishnu, Isha (Shiva) and Bhaskara (Surya, the Sun God);
Grihsthas worship all;
Brahmacharis worship Brahman;
Vanprasthis worship Arka (Surya, the Sun God);
Sanyasis worship Maheshwara (Shiva);
Bhutas worship Bhagvan Rudra (Shiva);
Kushmandas worship Vinayaka;
Prajapati Brahma is god for all;
But Men should worship Hari (Vishnu) after realizing that Rudra (Shiva) is no different from Vishnu. Else, Hari (Vishnu) will not destroy the enemies of the king.
sectarian Hindus (in modern times, this amounts to more or less just vaishnavites) only worship the god prescribed for their sect. non-sectarian Hindus may worship one or all of their grama devata - the guardian deity of their village or their kula devata - the deity worshipped by their clan or their ista devata - or chosen, cherished deity.
Most Hindus today worship multiple Gods who may include God-men like Sai Baba.
Some deities are worshiped for specialized reasons - garbarakshambigai to get pregnant and deliver the baby safely, the deities of some temples are worshiped to cure disease and so forth. And unless their is a sectarian clash all Hindus would go on pilgrimage when they get a chance and would worship the deities of 'marquee temples' like Puri Jagannath, Chidambaram Nataraja, tirupati venkatachalapati etc.
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As S K already said, 'non-sectarian hindus(largest percentage) worship multiple Gods', but additionally they also have firm and vocal beleive that "all Gods are basically one, so it does not matter, which God basically one chooses". Hence, you will see that:
(a) Same Hindu when goes to Vishno devi yatra, he will affirm Devi as his favurite/biggest/isht devtaa.
(b) But next year same person may be going on pilgrimage to Vrindavan during holi, and he will proclaim krishna/vishnu as biggest/isht-devtaa.
(c) During Ram lila, he may proclaim Ram as jai shri ram or during death procession of somebody else he may proclaim - "Ram naam satya hai".
This practical fluidity of switching of isht-devtaa of majority of hindus comes from a philosophical basis called Smartism ( https://en.wikipedia.org/wiki/Smarta_tradition ) and also the fact that Hindu religion is basically Omnist ( https://en.wikipedia.org/wiki/Omnism ) at core level, it beleives multiple paths are all correct simultaneously, so they have no problem switching Gods(even isht-devtaa) and paths almost at fluid level.
Also, in Gita , krishna has himself also supported sankhya yoga, which does not beleive in God per se. Many yogic tradition followers, beleive Shiva as adiyogi (and not as God). So for them even ishta-devta choice does not even arises.
But there are sectarian hindus which will stick to a particular God, depending upon which guru/tradition/clan they got initiated from/belong to. So his clan may already have a particular ishta-devtaa, and he will follow that God as isht-devtaa.
As rightly said in this comment, for Hindu & non-Hindu the worshipping happens based on Nature (swabhAva / Prakruti):
BG 17.2 - The Blessed Lord said That faith of the embodied beings, born of their own nature, is threefold-born of sattva, rajas and tamas. Hear about it.
BG 17.3 - O scion of the Bharata dynasty, the faith of all beings is in accordance with their minds. This person is made up of faith as the dominant factor. He is verily what his faith is.
BG 17.4 - Those with sattva, worship the deva-s; those having rajas, worship the ogres (yaksha) and monsters (rAkshasa); Other people possessed of tamas, worship ghosts and the spirits
In today's world, it's complicated to know who is praying whom due to various purpose underneath. e.g. if X prays to lord Krishna for money then actually X is praying "money" (Kubera). Through the particular deity one also gets results of their prayers.