Welcome over from the Dark Side! The fact that you have a tree in your yard doesn't change the advice (the Scott's products warn not to use near trees because it will kill them).
To get started, here is a link to Howard Garrett's Organic Program Philosophy
There is some elbow grease involved in the program - for example, now is a good time of year to pop those weeds with basal leaves out of the lawn, plants like thistle, many of the composite plants like dandelions, and some of the tough grasses like dallis grass, etc. Get them when they're tender and you can wear a pair of heavy gloves and pull them out carefully, or use a weed popper (the thing with a few nail-like spikes - they work, but sometimes leave a divot to replace).
This is the kind of thing I'm talking about:
For spreading weeds like henbit, and dichondra, you can spot spray with vinegar (10% pickling vinegar, the kind you get at grocery stores - sometimes hard to find in the city, I go out to small towns outside Fort Worth and find a David's Supermarket - that chain has it at about $2.75 per gallon). Spray it at full strength, and add a couple of tablespoons of orange oil if you want to make it stronger (shield the grass around it with a bit of cardboard as you spray). Be sure to empty and rinse out your sprayer or the vinegar will eat up the rubber gaskets).
Take a look at the Organic Guides
for a lot of information, and for specific questions about products or plants, look in the library of organic information. I've circled the two sections I mentioned here on the DirtDoctor home page image.
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