Looking for a service organization (non-religious) to join

Is there a Junior League near you?

League of Women Voters. Get qualified to register new voters. It’s nonpartisan - the goal is just to increase voter turnout.

I am working with Rotary International and they do tons of great work at home and across the world.

girl scouts. I both do training to support leaders, but also active in our local alumnae group.. we provide volunteers for events, have fun activities and raise money for capital projects. there are opportunities for weekly or just occasional volunteering.

I do Meals on Wheels, volunteer with the March of Dimes and help with a home for women who have children and have left abusive situations. In the past I have gone to schools and read to Kindergartens as well as taught Art in Education (a local program, I think, that volunteers/parents present a short 15 min age appropriate lesson on an important artist. They would give us the outline, then we would 'teach' the lesson. 

When I originally read your question, I was thinking you wanted a social club instead of just somewhere to volunteer.

This was going to be my suggestion too! 4-H is awesome and it is WAY more than farm animals and growing crops. My kids are city kids that just happen to live in the country. LOL!

They learned public speaking, photography, sewing, buymanship, cooking and more!

If there is a local Fisher House to you that can be a very worthwhile volunteer experience.

Local Humane society, Meals on Wheels and most of my mine is spent with The Lions Club, which supports a lot of various endeavors besides vision related

Fraternal Order of Eagles (Foe.com).

“People helping people.

Don’t know where you are located, but the Eagles are an International Order.

On a local level, I know people that volunteer for the summer season at our nationally ranked children’s zoo.

For the music lovers, my DH just signed up to volunteer at our Historic Theatre that features nationals productions from the Broadway stage, concerts of all musical formats, cinema, educational programming and a continued commitment to young people.

Another Meals on Wheels volunteer driver here and I also volunteer at a local seniors centre.

I also volunteer with both Girl Scouts and Boy Scouts! 

I would actually go about it in reverse. Figure out your skills and talents and the kind of things you'd want to be doing and how much time you'd like to spend, then figure out what organizations can use those.

Also FWIW, there are a lot of organizations that have their roots in a church or religion, but that aren't religious themselves. I'm involved in a large local charity that was begun out of a Catholic church and technically has a religious name, but isn't religions itself, and I'm aware of several others in our area that have similar types of stories. If you find a charity that seems worthwhile but a name or what you think you know puts you off, be sure to double check.