Clover is indeed one of the plants pictured, but there also appears to be Wild Violets, which are native and of value to pollinators (several butterfly species can use them as caterpillar food), though some gardeners consider them to be lawn weeds. If you aren't bothered by having some clover mixed-in with the lawn, you don't necessarily need to remove it or suppress it with herbicide before overseeding. "Microclover" is the subtype typically recommended for use in a mixed lawn, since it is less aggressive than "regular" white clover, though we can't tell from the photos which version the type present may be. If it was originally seeded with the lawn decades ago, it might be regular clover. The only way to remove it, if you wanted to do so, would be to either physically strip it (along with the sod it's mingled with) or treat it with a systemic herbicide that targets broadleaf weeds and not grasses (these are widely sold).

If you did opt to remove the clover-infiltrated sod with a sod cutter machine (rented yourself or used via a hired lawn company), then you could either lay new sod or put down seed, both of which should wait until around early autumn (or very late summer) for cool-season lawns like fescue. Sod will cost more than seed per square foot of coverage, but it will provide a shortcut to establishment, plus give you a weed-free start to the lawn that should last at least a couple years if the sod establishes well.

For any process, having a laboratory soil test performed is the best first step, because this will tell you if lime is even needed to adjust pH, plus whether any nutrients are deficient, which informs what choice of maintenance fertilizer you'd be using. (Phosphorus-free, for example, or not.) We can help to interpret test results if desired. If you already know you need to add lime, then that can be done first, yes (maybe around a month prior to seeding, though it can also be done just about any time); lab test results will guide you as to how much lime to apply per given area of lawn.

Core-aerating is done just prior to seeding, so the exposed soil is ready for seed sowing, since good seed-to-soil contact greatly improves germination rates. Aerating might help to incorporate lime and/or fertilizer, but those items could also be applied post-seeding as well...the timing isn't critical, and it would make more sense to fertilizer after seed is present than before, since you don't want rain to start releasing some of those nutrients (like nitrogen, which can be water-soluble) before the grass roots can start to make use of them. For fall lawn fertilization, you'd be making a second application later in autumn anyway, as the grass is becoming established.

Miri