I have a custom list in a webpart. I need to search that list based on some filter conditions. How can we do that using visual studio code ?
I'm going to assume, based on your question, that you are new to SharePoint development. So, the first step is to spend some time to understand the fundamentals - here's a video that will help you get started. It's for SharePoint 2010 but most of the material applies to SharePoint 2013 as well.
Then you want to take a look at how to build web parts:
Then you can move on to what you need to do to query the list. In SharePoint, this is done using a CAML query.