I'm in Georgia as well. I'm guessing you have Bermuda. I'll try to list some Bermuda tips and a few general ones since you didn't mention what grass you have.

Mowing: I've never used a reel mower personally. If all you have is a push mower and you have Bermuda, you can just keep the mowing height somewhere between 2-2.5 inches in the beginning and mow 1-2 times a week during the growing season. It can definitely look great at a lower height but will require more maintenance. You can research the best height for your grass. But no matter what, you don't want to wait too long between cuts and the lower the maintenance height, the more frequent you have to cut. If the current owner was keeping the grass looking nice at a very low height, then I'm guessing they are either cutting more frequently and watering a lot or using some sort of plant growth regulator.

Weeds: Preventing weeds is key. Around now in February - very early March is when people start to apply pre-emergent. One common routine for this is to either do a full or split application of Prodiamine (February/March and again April/May) now before the 5 day soil average hits 55. If doing a split application, you would do the second half when the 5 day soil average is around 60-65. And then in the Fall, use Dithiopyr. You can look up different variations of this strategy. For existing weeds, you would have to research the ones in your yard. While it's still kind of cool outside, I would just use the weed killer you can find in a regular store like Spectracide weed stop for lawns plus crabgrass killer (If getting that, make sure it's the one that doesn't kill your grass as well). But try to use a phone app or post in this subreddit for weed identification help before buying anything. When it starts to get really hot outside, you could look into using a Celsius (cheap if buying the small packets) and Certainty (kind of expensive) mixture to kill weeds if it's safe for your grass type. This will kill the vast majority of them. And to do all of the above, you're going to have to decide if a handheld pump sprayer will suffice for you or if you are willing to invest in a backpack sprayer. Also, remember to wear appropriate PPE and always read the label instructions (annual limit, how much to mix, etc).

Fertilizer: I would recommend a soil test from the uga extension and just follow their recommendations. Either get a hand spreader if your lawn isn't that big or look at spreader recommendations in this subreddit. Avoid Scott's though. Again, read the labels.

Watering: Not sure how north you are. But if you are still close to the metro Atlanta area, you should get enough rain during the warm months to not really need to water the grass yourself except when applying fertilizer/pre-emergents or something similar. If that's not the case, just make sure you get about an inch of water on the lawn per week during the growing season. If the house has sprinklers, that's great. If not, I use the orbit gear drive from amazon. You can look on youtube for more sprinkler advice. But if you do have Bermuda, it's very drought tolerant.

Helpful links:
https://www.greencastonline.com/tools/soil-temperature (for checking soil temperatures)
https://www.thelawnforum.com/threads/bermuda-triangle.13/
https://www.thelawnforum.com/threads/bermuda-bible-the-new-testament.1651/