For what you're planning you really don't need much.
Unless you are going off-trail or doing something extreme you don't need special boots. Most of the main trails in the National Parks are wilderness freeways, wide and well maintained, worn from travel. Tennis shoes are fine. At most, light hikers. You shouldn't need ankle support unless you have especially weak ankles, and even then some hightop light hikers should be sufficient.
If you do buy light hikers, brand matters less than fit. Go someplace you can try them on to ensure correct fit (no rubbing, constriction points, not too loose, etc.). No need to buy super expensive, around $100 should be fine. Or less if on sale.
Do not buy heavy duty hiking boots. They are expensive, take forever to break in, and are overkill for your plans. These type of boots are for those hiking long distances, and/or doing lots of off trail travel. They are also heavy, which takes a toll.
As for other gear: Daypack (small backpack), sun hat, water bottle, and camera. I prefer long sleeves (even in desert) and pants just to keep the bugs/sun/dirt off, though this is largely a matter of personal preference. Recommend wool or synthetic material (pretty much anything except cotton). Weather in any of the National Parks can be unpredictable and highly variable, so good idea to pack along a lightweight waterproof and windproof jacket (if you already have one) or a waterproof rain poncho (run about $30).
If you're getting off of the main trails/attractions and further into the interior of parks then you need a few more things: topo map and compass (and know how to use them), water filter, first aid kit, warm layer and some extra food and perhaps a lightweight tarp in case you have an unplanned bivouac. But I don't get the sense that this is the type of hiking you have planned.