I think it depends a lot on your location as well as the nature of the items you're buying/selling.
When buying: I gravitate toward the highly detailed ads with lots of pictures, since these people put in time and effort they are more likely to be legit (or more likely to be scammers, ha). I don't bother with the one-line ads with one blurry picture. Next, I always call the person for my first contact, and ask a lot of detailed questions. You can usually tell a lot by a person's tone of voice and how knowledgeable they are about the item they're selling. Once I decide to come look at an item, I'll work out the rest of the logistics by text message and maybe a call as I get close.
When selling: I make an ad that I would myself pay attention to (see above). I'll publish my location as the nearest major intersection, so they know what neighborhood to come to. I'll screen responses, and answer phone calls before texts, again judging by tone of voice how reliable/flaky they are. If they want to come take a look, I'll tell them to call me when they arrive at the cross street in the ad and I'll give final directions over the phone from there. This weeds out a LOT of flakes.
A few times I've met at some arranged public place instead of one of our residences, but the majority of my CL transactions have been at my house or the other party's.
Payment: Cash or paypal, never checks. I used PayPal for a large CL transaction recently and we both loved it. Allowed me to make a low risk deposit so the seller would hold the item until I was ready to come back and pick it up (and time for him to write up a bill of sale); afforded all the standard PayPal security and protections; allowed me to use my CC for a ton of airmiles; avoided the hassle and risk of large quantities of cash. I know PayPal takes a cut, but I feel it was well worth it and I'll use it as a seller next time I sell something big.
Haggling: When selling - It depends. If it is something that I would otherwise be donating at the next opportunity, I'll accept the first offer I get. I have unloaded several kids bikes over the years for $20 or $40 each, and donated the ones that didn't sell. Most things I put on CL fall in this category, so I don't set a really firm price. When buying: I consider the ask price before I even bother calling. If I think something is priced too high, I'll make them consider a reduced offer over the phone before I even come over to take a look. If it's appropriately priced for CL (i.e. 25-50% of what I think it would cost new) then I usually pay asking price. It depends, a lot, upon the circumstances. I've sold much more than I've bought on CL, and most of my sales were to de-clutter, so I don't drive a hard bargain.
As a seller I will not deliver something to a buyer, nor would I expect the reverse. Unless a cash/paypal deposit has already been made.