The two best shore dive spots that are accessible most any day of the year are Pos Chiquito and Mangel Halto. (Actually they are the same reef.) These two also make for a nice drift dive entering at one and exiting at the other with a short 5 minute walk down the road to your car.
I haven't had a chance to stop by either lately, but there's usually a bleach bottle underwater tied to a line at 40 feet at both places that marks your exit point. The reef is nice and starts at around 20 feet, gradually slopes to 40 and then is a steeper slope past 100+' where it flattens out into sand. The entry at Pos Chiquito is easy. Just swim out through a small surge staying close to the rocks on your left 'till you're in 20 feet of water, about 300-400' and drop down and make your way out over the edge. If you're heading out on your back, keep the satellite dish lined up with the edge of the dock. This is a nice easy nite dive too. Mangel Halto is a little more difficult. You have a long walk through the water to the dropoff of "the bowl" which is about 60 feet deep, round with a sand bottom and is still inside the barrier wall. You then swim out a short ways to a marker where there is a channel running through the barrier wall that will get you outside to the reef. This channel is 5 - 10' deep and the bottom is lined with fire coral. There's usually some surge, nothing special, but you want to watch were you go so you don't hit the coral. From that point, there's a gradual sloping reef till you get to the wall. If you drop down inside the bowl to swim to the marker and then the channel, take a compass heading or you'll be swimming round in circles inside the bowl and there's not much to see there.
Another nice dive is Saveneta. The reef is about the same, but the entry can be a little more difficult to spot. The reef comes shallow to about 1-2' and forms a natural break wall that you have to swim past to get to the dive site. If there's waves, as there usually are, there will be a small break in the surf where it's about 5' deep, that's your entry point.
Most of the sites listed up north on ShoreDive are really snorkeling sites with sparse reef and not a lot of wildlife. The three sites I listed are worthy of multiple dives in heading in both directions from the entry point. ShoreDive gives a good description of the sites along with pictures, but unless you have a GPS, directions on how to find the site are sparse. Whoever you choose to get your air from should be able to give you directions.
Baby Beach used to be a really nice site, but Hurricane Ivan threw all the coral up onto the shore. Jads' dive shop is located right there. He can give you info on entry and exit since it's changed since Ivan.
That said, Aruba isn't entirely surrounded by reef like Curacao or Bonaire and where it is, there's usually perimiter islands or sandbars between the mainland and the reef, so a lot of the dive sites are only accessible by boat. The prettiest reef that the dive operators go to is in the Bacadera / Balashi area and is almost as far south as Mangel Halto. There's some great wrecks for the casual wreck diver that are really must sees if you come down here. The Antilla is a 425 WWII German supply boat sitting in 55' of water. The Jane C is in 60' to 90' of water and is a nice little 170' freighter. There's a new, a little over 2 years old, DC4 airplane sitting upright on a slope in 85 - 55'. For seeing wildlife, the best spot on the island is the Pedernales wreck which is just a lot of metal underwater as it's remains were blown to pieces by the Dutch Marines for target practice. The site is only 25' deep, but pretty much any critters you'll see in Aruba you'll see on every trip to this site. This is a real nice night dive too. Unfortunately, you'll need a boat to get to each of these wrecks.
As far as dive shops, SEA Fly and Dive is a good shop. I have nothing particular to say bad about Mermaid listed above. Native Divers is nice because you'll probably be by yourself and he'll take you wherever you want as opposed to following a daily schedule. My favorite was always Unique Sports. I've dove with just about everyone on the island and started doing all my boat dives with them. I'm slightly biased now since I DM for them on weekends when I can, but I made my decision long before I was affiliated with them. Now they have bought Pelican's dive operation and are the largest operator on the island. One nice thing they'll do that most of the other shops here won't is they'll let you run an open ended dive package instead of buying a set amount and scheduling them for particular days. Then at the end, however many dives you did, that's the rate you get. I'm sure that they'll give you a break on shore dive air if you do some boat dives with them. If you want Nitrox, there's only one guy on the island and everyone else gets it from him. This isn't really an environment where EAN is going to give you a lot of advantages though.