Here are some Android programs I use frequently, plus some of my favorite games. (Sorry if I duplicate some suggestions!)
Battery Widget Reborn (beta) (free, has a paid stable version). I like having the perma-notification, which I have set to show "empty at $TIME".
Google Chrome (free). Hey, it's a solid browser. Also recommended: ALL the Google apps, particularly Quickoffice.
Firefox (free). A little slow on older hardware, compared to Chrome, but still solid.
Fing (free). A neat set of networking tools. I fire this up a lot at home when I want to ssh into something, and can't remember each device's IP. (Why, yes. I am lazy.
Hacker's Keyboard (free). A quintessential keyboard if you regularly use your device as an ssh client.
K-9 Mail (free). An open-source mail client. Great if you don't like the stock e-mail app and don't rely on Gmail.
Retroarch (free). (Best with: MOGA, Wii Remote with Classic Controller) A great all-in-one emulation app.
Total Commander (free). If you're serious about file management, I can't recommend this app enough.
Gunman Clive (paid). (Best with: Wii Remote or MOGA) It's a lightweight Mega Man! 20 levels of mostly-brown fun.
Osmos (paid, demo available). (Best with: just a tablet) A beautiful, zen-like puzzle game. Gets a bit actiony at times, but patience is always key.
Riptide GP 2 (paid, optional IAP) (best with: just a phone, or MOGA). I missed out on classics like WaveRacer 64, so this is the game that got me into water racers. Gets a bit level-grindy in difficulty. IAP for in-game currency, but I cleared the game without it.
Sonic 1, Sonic 2, and Sonic CD. (paid) (Best with: Wii Remote or MOGA). Brilliant ports of their source games; these all use Christian "Taxman" Whitehead's "Retro Engine". Sonic 2 has a "restored" Hidden Palace and smoother Special Stage.
Super Hexagon. (paid) (Best with: just a phone or tablet, or MOGA shoulder buttons) It's Super Hexagon.
World of Goo (paid, demo available). (Best with: just a tablet) I find the game more enjoyable on a touchscreen.
I'll edit this with additional favorites later.