Although I wouldn't say I was religious, the majority of festivals I've seen have been religious ones.
In May this year I went to Echternach, Luxembourg (on a day trip from the German city of Trier) with my mother and two sisters to see (and take part in) the famous Hopping Procession on Whit Tuesday. It was a fabulous experience, not dampened by the rain.
While in Mexico City last year I went to the district of Iztapalapa to see the Good Friday part of the famous passion play re-enacted there every year. I've also seen the Good Friday processions in Quito, Ecuador, as well as celebrations for Easter Sunday. I've seen the midnight Bulgarian Orthodox Easter celebrations in Bulgaria, too.
When I was in Ecuador, I was also present during Carneval, so I saw lots of foam-squirting and water-throwing in action.
When in Barcelona for Christmas just before the pandemic, I had the incredible experience of Midnight Mass at the basilica of Santa Maria del Mar. The whole thing was amazing, but special mention to a procession at the start, of hooded people with candles processing to a haunting, beautiful tune played by Catalan bagpipes. It was incredibly atmospheric; something I'll remember for the rest of my life.
I have seen celebrations for non-religious local or national holidays as well, though they've tended to be ones where I was already in the country, rather than ones I went to the country specifically to see. For example, when living and working in Bulgaria, I saw processions for Veliko Tarnovo Day and Bulgarian Independence Day. When in Ecuador for the same reasons, I went to Cuenca for a long weekend partly due to the Independence of Cuenca and All Souls' Day celebrations. I was based in Guayaquil and so I also saw the Independence of Guayaquil processions there. Elsewhere in South America, in Cuzco, Peru, I was able to see Corpus Christi processions over two days in June (this was even longer ago than Ecuador, though). Speaking of Corpus Christi processions, I've also seen two of these in Lisbon, in 2012 and 2022.
Years later, when I was Maribor, Slovenia for only one full day, it turned out that it was Prešeren Day, a public holiday celebrating Slovene language and culture - I hadn't known this in advance though. In Amsterdam for a week in 2013, it coincided with the last Queen's Day before she abdicated, so I saw the public celebrations for that. When in Japan a couple of years ago I was there at Christmas (a secular event for the vast majority) and New Year. I was also there for a couple of the autumn public holidays, but I don't remember actually seeing any celebrations for them apart from people having the day off. In 2008-09 I was in Australia between November and February and so I saw New Year's Eve fireworks (in Cairns) and the Chinese New Year parade in Sydney.
I remember once on a school trip to the WWI battlefields of France and Belgium, we were staying in Nieuwpoort, Belgium for at least a night, and my sisters and I went for a wander along the seafront late one evening and saw something unexpected - fireworks and lots of ship foghorns sounding. Since then I've tried looking things up, but none of the Belgian public holidays match up with it. It wasn't an occasion that our trip was planned around at all, it was quite unexpected and something we only saw because we'd wandered over during some free time. Even 20 years later, it's remained a bit of a mystery for us; we still wonder what the occasion was that we saw that evening.
