My employer mentioned sponsoring me for a work visa in the US but I am not sure how the process works. Do I have to apply myself or does the company handle everything

In most employment based visa categories the employer starts the process by filing a petition with US immigration authorities on your behalf. Depending on the visa type, the company may also need to complete steps such as a labor condition application or labor certification before the petition is filed. After the petition is approved, you would typically apply for the visa at a US consulate or change status if you are already in the country.