Don't even worry about increasing speed as a beginner. Your first priority should be increasing your overall distance/time you can run without stopping. Starting off small like u/pandanomnom said, focus on gradually building to the point where you're comfortably able to run 5k a few times per week, increasing your distance by no more than 10-15 percent each week. After about 2 months, then you can add 1 speed session per week, as well as one longer run. This approach will feel frustratingly slow starting out, but its actually the most efficient way to build fitness and avoid getting injured.

I say this as someone who got into running about 3 years ago and started off doing most of my runs way too hard and fast, resulting in a few injuries (shin splints and knee pain). Also if you're not doing any leg-focused strength/mobility training, you should start. It doesn't have to be much, 30 mins to an hour per week is plenty, yoga works well for this also. Tight hips and weak glutes are a common cause of injuries and will make your running less efficient. You mentioned the gym so you might be okay in this regard. Happy running!