I'd be careful looking for jobs while at work, especially on a work computer. That's a great way to get let go if they notice you doing it.
As far as the search itself, you currently have more leeway because you've got a job - you don't need something tomorrow, so you can take more time. Start by talking to any friends or friendly acquaintances you have in the same field you're looking to work in. Ask about their current workplaces, their former workplaces, etc. to see if any stand out as places you'd also like to work.
Once you've spoken to any people you know, consider companies you've heard good things about. Check sites that share employee reviews like Glassdoor, look them up on LinkedIn, etc. Your goal should be to get a sense of average employee tenure as well as potential pain points. Two big things to keep in mind as you do this:
- No workplace is perfect. Every job is going to have its flaws, so you're looking for one with flaws you're okay dealing with in the mid- to long-term. If you're looking for perfection, you'll never find it.
- Employees who post reviews tend to feel very strongly about their employer, and those feelings trend negative. That doesn't mean you shouldn't trust the reviews, but it does mean you should take them with a large pinch of salt and look for trends rather than specifics.
Once you've found companies that you think might be a good fit, try reaching out to employees at those companies via LinkedIn (or some other site where they congregate) and asking for a brief chat. You don't want to take up a lot of their time, but a quick "I'm thinking of applying to X and would like to hear what working there is like before I do" should hopefully get you some firsthand information. It may also get you a referral, which is a big bonus when applying.
Even after all of this, it's important to remember that you still probably won't have all of the information you'd like to be able to make a confident decision. That's unfortunately just not how it works most of the time, which is why so many people choose the devil they know and stay instead of risking going elsewhere.
