First of all, congratulations on the new home!
There is just nothing out there that will do it all well: fertilize, kill weeds, insects, fungus, etc. To get a good result you will have to treat each separately. A few guys I follow on YouTube for advice are Ryan Knorr, GCI Turf, Lawn Tools, Lawn Care Nut, Lawn Ginga, Lawn Rebel, and The Lawn Whisperer.
I've been using the YardMastery fert and so far really like it. You have to order online but it's good stuff.
My fert plan this season on my bermuda and zoysia:
April and May: Flagship 24-0-6
June: Double Dark with 6% iron 16-0-0
July: Stress Blend 7-0-20
August: close out the season with Flagship 24-0-6
As others have said, super important to lay down a preemergent in the middle or at the end of February before crab grass and weeds start. I spray it with prodiamine 65WDG (water soluble granules) in February at the full application rate for my first preemergent application. At beginning of March I spray it with Demension (dithiopyr) at the full application rate. The two preemergents target different zones in the soil.
I spot spray weeds before it gets too hot with a mix of Speed Zone or T Zone and Blindside. In the winter, WHEN THE GRASS IS DORMANT!, I will spot spray with 43% glyphosate. When nutsedge pops up I spray it with Dismiss. Dismiss is expensive but it whips Nutsedge's ass. You'll see it begin to yellow 24 hours after the application.
In March I apply Acelepryn G or Grub X for grubs and insects.
If I begin to see bermuda seed heads pop up I apply T-NEX PGR (plant growth regulator). PGR also keeps your lawn from growing tall quickly, meaning you mow less frequently. It slows the grass growth allowing the grass to use more of its energy towards the root system and overall health (thickness?) of the grass.
I also highly suggest getting a soil sample kit and sending a soil sample in for an analysis. This will help you determine what your particular yard needs.
My plan is pretty involved and I really enjoy yard work so this may not be for you. So far my lawn is three shades darker than both of my neighbors and I have very few weeds. One of my neighbor's yards is solid crabgrass and you can clearly see the line I sprayed heavy with prodiamine between our two lawns..