Your first step should be getting a comprehensive soil test, so you can apply the correct ammendments. Core aeration is usually only necessary when soil is heavily compacted. Doing it 3 or 4 times is overkill, once is enough. Most soils are low in organic matter, a soil test will confirm this, but applying some organics is usually always a good idea.

Your soil needs organic matter, lots. What you want is between 5 and 8 percent residual organic matter (humus) in your soil. Since the soil pH is very high the sulfur is needed to get that back down, optimal for most all plants is 6.2 to 6.8. Sulfur should be available at any good garden shop, although for how much you need you may want to seek out a farm supply store. Before adding any "fertilizer" I would get the compost down as well as the sulfur, see if that is enough compost, add more if necessary, and then seed the grass. Since you will be tilling in this sulfur and compost there should not be any need to aerate that soil any further.

**Try gathering the free hay sweepings from a feed store. I've done it lots of times as they use a blower or broom/dustpan to clean up at the end of the day, so it's a mixture of hays inside a dumpster. I wear coveralls, gloves, dustmask, etc. & bring my own bags & containers then clean up when I'm done. As long as it's not your top layer there is little sprouting of seeds. **The beauty of the lasagna method is that you can collect your materials ahead of time, then redo the bed in one day. -dig your plants & set aside on a tarp or into cardboard boxes in the shade (keep rootballs intact & shovel fulls rather than bareroot) -cardboard, then layers -top with good compost (weed free) -lay out your plants, part the layers & plant -mulch them -keep watered for at least 2 weeks until plants established again We did this very thing this spring for my hubby's parents' place where the sod was nearly impossible to remove from the bed as grandpa dug a hole in sod to plant instead of making a garden bed. You could barely get the shovel or edger into it. We didn't have our trailer to haul manure or compost, so we ordered a delivery of local mulch (chipped from old piles of tree scrap). Used coffee grounds from a shop where they pile it outside for you to scoop & haul, our kitchen waste from the week, some dried lawn clippings, some plant prunings mown over, etc. None of the layers were very thick or too much green, so it didn't heat up. I've done it several times here as well using partially composted manures as the bottom layer topped with the hay sweepings. I've planted right in them despite warnings about letting it compost fully 1st. We did one in the backyard this spring. Here's a pic from a few weeks ago before the daylilies started blooming here. The buildings behind are chicken, duck & rabbit coops, so this bed was quick to make from cleaning out the coops! We made another one the long length of the front yard. It surprised the next door neighbor who asked hubby, "Did you expand the garden?" who replied, "Yes, a little bit." Neighbor said, "Looks like a lot!" It is amazing how quickly you can do it!

Every nursery has a different overstocked inventory they want to sell you, don't listen to them--Get a soil test from logan labs & post the results here for amendment recommendation. There's some other guys on this site who live in California & finally got rid of their under-performing marathon, but it depends heavily on your location's micro-climate. Fescue is a shade grass that can't tolerate traffic well, but the nurseries & sod farms in CA love to sell it to everyone so you keep coming back each year for more useless amendments & sod replacements. If you can get it from a farm locally, St. Augustine would be my choice unless I lived in the desert areas, in which case I would chose bermuda. Seal Beach doesn't sound like a desert to me though... For shrubs, find something native so you don't have to worry about it crapping out when the next round of water restrictions hit. If you don't like anything native, tecomaria has nice red flowers, vitex are intense blue, or you can get Ixora in just about any color you like.