If you are using Klaviyo's default Welcome Series, double-check that it is triggered by contacts being added to the correct list. Klaviyo's default Welcome Series is triggered when a new contact is added to the default Newsletter list. It may be the case that your signup forms are connected to a different list, which is why your Welcome Series is not triggering.
To verify that your Welcome Series is connected to the correct list, click on it in the Flows tab.
If you're using Klaviyo signup forms to collect new subscribers, verify that these forms are pointing to the correct list. You can do this by heading to the Signup Forms tab, finding your live forms, and clicking on the submit button.
Last, if you have newsletter signups synced through your ecommerce integration, head to the Integrations tab, toggle to Enabled Integrations, and find your ecommerce platform. Here, you will be able to see which list new signups are added to. This should match the list that is triggering your Welcome Series.