I want to make a form which contains one select field (dates) and multiple checkbox fields (teams), so when I select a date or choose a team, automatically makes a query a database, filtering by the selected date and teams. However, I don't know where I have to start, so any advise would be greatful. Thank you in advance.
RSForms Pro is very powerful and can do this. There are a few steps to it. When you make your fields, you would auto-populate the available teams by using RSForms built in shortcode for the query. Then you actually post the data and grab it back.
You can create a search form and add some custom code to query the database and display the results using many of the forms extensions in the Joomla Extensions Database e.g. ChronoForms.