So a while back I had been praising FarSight's pinball collections from a few years ago. Now the developer is back with Pinball Arcade. This is half repackaging of their old work and half new tables, sold ala carte. This is for every platform imaginable, from PSN to XBLA to your iOS/Android phone or tablet. I think there's even a Vita version.
The collections are still the best value because the old Williams Collection is so cheap now, but I've bought every Pinball Arcade table on my iPhone for the convenience and to play the new tables as soon as I can. This is definitely my phone game. I recommend you pick up Williams Collection for cheap and see if you like it, then go to this.
The collection is really deep at this point, and since you buy tables individually I thought I would make a recommendation post with a beginner in mind. Keep in mind the iOS/Android versions are most current, Xbox version is way behind on tables, and I believe the Sony versions are slightly less behind. As I write this, Black Knight and Creature from the Black Lagoon just came out.
Arabian Nights is free, so you'll be playing that. I'm not crazy about this table because of all the obstructions and the lousy score balance. As far as I can tell you can ignore everything but hitting the lamp over and over again, which gives you way more points than anything else you can do on the board. Even the reward for actually completing the table kind of pales in comparison.
Now Theater of Magic is a great one to start on: it's easy to play, easy to score on (and you get a million for practically anything you do), but there's a lot of depth and room for skill in the table rules. I've probably put the most time into this one. Cirqus Voltaire is also pretty forgiving and is a very unusual table. I'd love to see one of these in real life.
I haven't had Creature From the Black Lagoon for very long, but it's growing on me fast and it might be one of my overall favorites in the set. The theme is brilliant (it's a meta-game about going to a 50s drive-through to see the movie Creature, which you have to earn by getting into multiball), the difficulty is medium, the table's addicting. Try it out.
Monster Bash is probably the easiest table to beat: I got to the wizard mode the second time I played it. I guess it's technically a good beginner table, but I find it a bit too easy, and the theme is really obnoxious (the Universal Monsters start a rock band in the late '90s; catchphrases from The Mask are used). It's kind of like a much dumber version of Medieval Madness. I would outright avoid Bride of Pinbot, which seems like fun until you realize that there is literally no reason to do anything but shoot the left lane over and over again. Just a bore. Of course, it's packed with Medieval Madness, so...
Speaking of which, Medieval Madness is a beloved table, but it's also kind of perfectionist and brutal: missed shots are usually punished by death. Recommended, very basic, very addictive, but not for your first. Ripley's is also a great expert-mode machine with tons of crazy systems and secrets. Both of these are tough as hell, so be prepared.
Funhouse is uneven but I have a soft spot for it: everything is about earning one high-value jackpot, which is one of the hardest shots on the table in a stressful situation (multiball). The table is viciously designed, but it gives you so many extra balls that it kind of evens out. I imagine if you got really good at this table you'd be playing it for an hour or so on one credit.
Don't overlook the vintage tables: you'll be missing out on a whole other genre. Gorgar is extremely simple, one of the coolest-looking tables, and extremely fun. Just hit the drop targets and don't die. Plus it's got that amazing voice. YOU. BEAT. ME. This one's packed with Monster Bash, so I guess you'll get that one too, you poor buyer. Black Knight has a few more bells and whistles, like multiball and a score multiplier (whoa man), and it's great fun in the same vein. Black Hole is a really complex vintage table and pretty cruel. Look up how to get extra balls in the instructions (it's the drop panels) and it goes from painful to actually kinda fun.