I’ve started/rehabbed a few lawns in this area pretty successfully (once I rehabbed my own the in-laws/friends suddently felt the need to enlist my help). To be very honest, starting a new lawn this time of year is an uphill battle. If temperatures get high in late May/early June like they did last year, your new seedlings will not have time to establish roots and will get fried. I would suggest taking the next few months to get your soil in shape — till and compost heavily to break it down, get rid of any weeds that may be populating your current soil, and see if you can get to a good pH for lawn growth — internet has plenty of info.
If you do want to start a lawn now and can afford it, laying sod may be your best bet. If you want to seed, a blend of ryegrass with fescue and/or bluegrass is good. The ryegrass is a bit heartier and less succeptable to heat. Most of the commerical brands (Scott, Vigoro, Pennington, etc) are hit-or-miss in my experience. Best bet is a local garden center if you want to avoid the commerical brands.
For your first feeding, you want a quick-release fertilizer with a high phosphorus number to promote root growth. Most fertilizers are labeled with a three-number label (e.g. 24-8-24), the middle number is phosphorus. Look for something around 14-18-14 for the first feeding. It’s a bit of a drive, but Merrifield Garden in Fairfax has an in-house 14-18-14 I’ve used with success before.
Four weeks later, use a slow-release fertilizer