Honestly, I am surprised at the question. This is how the majority of tours work. There are thousands of companies offer tour all around the world. All you have to do is Google Search a desired destination plus the word tour. It can be as specific or a general area are you like:
- Europe tours
- Spain tours
- Madrid tours
Tours where you are not travelling with other people are generally called private tours which are much more expensive because costs are not shared between so many people. Also, when looking at tours, group tours have specific dates stated since people must travel together. Highly popular tours can run each week and less popular (or very expensive) ones can have only one or two dates per year.
Most group tours do not include airfare to where the tour starts and from where it ends. So try to arrive one or two days before not to miss a tour which goes out far the very first day. For example, tours to Machu Picchu usually pick up participants in Lima and fly the next day to Cusco, so if your flight gets delayed, then it is difficult to catch up.
Since you stated your goal is to meet people, you can do improve your odds of encountering people you can relate to by looking for a specialty tour of your interests. Example of those are photography tours, culinary tours, adventure tours, cycling tours, etc. They even have singles tours where people who participate are usually single.
Some companies even have tours where the goal is to meet locals. Usually they include stays with local families, rather than hotels. This can be fascinating and they can teach you local cooking, crafts among other things. If that interests you search for local living or home stay tours.