You can easily go to a beach on Aruba all by yourself. My favorite is Arashi Beach, just past Malmok and very close to the California Lighthouse. It's right off of a residential area, with virtually no tourists at all. There are no services at this beach -- just beautiful sand and palapas spread out along the beach. Best way to get here is via a cab. Get there fairly early to make sure you can get a palapa, unless you want full sun all day. Note that there are no chairs there either, literally just palapas. So, you'll want to use your towels in the sand and use rocks to hold down the corners of your towels from the wind. You'll feel like a local!
We've spent time on land in Aruba, and what we did was we picked up sandwiches, grabbed a cooler and made a nice beach day for ourselves at Arashi. If you need services, you'll want to head to a more touristy area such as Palm Beach or Eagle Beach. The Radisson on Palm Beach offers a day pass, and Amsterdam Manor on Eagle Beach offers one as well. If you opt for one of those beaches, best way to get there is via the bus. They run frequently, are inexpensive, and safe. It's a really easy way to get around town, and even go beach-hopping if you'd like.
Moomba Beach isn't bad. They have a palapa bar and restaurant there, and the only reason we didn't spend a lot of time there was because it was fairly loud with thumping music. It's just not our style. Moomba Beach itself is pretty small, but they do have services and activities there, so you won't get bored. About all I did there was order a Baileys for my coffee and have my picture taken in front of the sign.
The only beach we really didn't like was Baby Beach. WAAAAY TOO MANY PEOPLE! Aruba is our favorite Caribbean island. Enjoy your time there!