I did a post like this when I first got my Evo 4G a year ago on here and on Android Forums. However, I believe some things have changed now, so, I will post my recommendations.
*Handcent -- By far, the best text messaging app you will ever use! Its light years ahead of the stock messaging app (even though the Sense 3.0 is improved over the stock) You can customize just about everything about your messaging, plus lock out certain text conversations. It even gives you a qyuick reply option too. You MUST give it a spin. Oh, and did I mention its free?
*Google Voice - You will want to use this over the Sprint's visual Voicemail app. For one, its free. The app sets Google Voice as your phone's Voicemail. You get free text transcribing of each of your messages and you have the option use GV to call out (so that your GV number shows in caller ID, rather than your actual number) and you can manage your messages online. Google Voice offers all Sprint subscribers to integrate their GV number with their sprint service. I WOULD NOT do this until you have read other threads on the issue.
*Google Goggles - Cool app that lets you take a picture of something and do an image search on it. It works great with pictures of text (even in other languages). Its OK with paintings and buildings. Does not work on pets, animals and people.
*ShopSavvy - This app scans bar codes using the phone's camera, tells you the product you scanned and tells you the nearest location to purchase based of your GPS position, along with the prices. Google Goggles scans bar codes too, but this is a nice stand alone app. Others use Barcode Scanner, but I find this one to be easier and better.
*Pandora - Who doesn't have Pandora? I guess I should ask who doesn't have a music streaming app, because not everyone uses Pandora. I do, and its great. give it a try.
*Shazaam - Nice app that listens to music playing in the background for 15 seconds and pulls up the name of the song, artist year and album, plus an option to buy from Amazon if available. The app is very useful. It even hears songs playing in the distance. There is another app, (I forget the name, but I'm sure others can help you out with that one) that does the same thing, except it allows you to hum the the song you're looking for. YMMV, but I hear Shazaam is better at finding the song, in general, just does not work with your own hums.
*Pulse News Reader - Quite simply, the best (visual) news reader app out there right now. Give it a spin.
*Compass - Nice app that makes use of your phone's built in compass. Can't go wrong here. There's even a view that uses GPS as well, to show how fast you're going and compares GPS Data to your Compass Data.
*Angry Birds - Because everyone has this on their Android phones or iPhones. If you've never played, its quite the addictive game and will take up your time for a while, that is until it gets boring and repetitive. If you have kids, they'll love it.
*MP3 Music Download - Someone told me about this app when I first asked the same question you did, so I feel obligated to pay it forward. I will not get into what this app does. You can find this out for yourself.
*Google Books/Kindle Books/ Nook - I know, I know. There's an ebook reader included on this phone. But give these a try. You'll especially want the Kindle app if you have a Kindle, or know someone close to you with one. You could share books.
*LAYAR - This is a one of those reality overlay apps that uses your phones camera to overlay images and info of whatever you are looking at. Its pretty cool.
*Hide It Pro - Sweet App that that
hides videos and photos from the main gallery viewer. So, if there are any pics or videos (i.e. porn) that you would like to keep on your phone, but would prefer that your kiddies or Girlfriend NOT stumble upon them while browsing through your phone, this is the app that you want. In your app list, it installs under the name "Audio Manager" and when you open the app, it indeed gives you several option for controlling your phones audio settings. There is a password protected back door that you use to get access to the files you've hidden. Very useful app.
*Speedtest - Nice app that tests the current speed of your data network, regardless of if your on 1.xx, 3G, 4G or Wifi.
*ChrometoPhone - Nice app from Google that lets you install apps directly from your chrome web browser, send web page links, pictures, and map directions to your phone, without your phone being directly connected to your PC!
*Google Music - I only recommend this app if you have a Google Music account. If you do not have one, go to music.google.com. to sign up for an invite. It is invite only for the moment, so it will take a while before you get that email saying you've been invited. It could take anywhere from a week to a few months. Google Music syncs your songs (up to 20,000 songs) stored on your computer to an online cloud account, giving you access to these songs from anywhere. Installing the app on your phone gives you access to these songs and streams them from the cloud.
*Tango - another video chat app to use besides Qik. Many people with Android phones and iPhones with Front facing cameras use Tango. Its pretty good and performs better than Qik.
*Evernote - Stores quick notes to a free online account, giving you access to them from a computer online, or your phone. You can record audio, photos and text notes.
*Handycalc - Best free scientific calculator in the market, IMHO. You should try it, if math is your thing.
ES File Explorer - Nice app that does what it says. Gives yo access to the file and folder structure on your device. There are many apps that do this, but this is the one I prefer for its many features.
*SMS Backup pro - Backs up your text messages, to a gmail account of your choosing. never loose that crazy text from 2 years ago. Useful for freeing up space on your phone.
*Weatherbug - HTC's weather app doesn't give you enough info? (Hint: No, it doesn't). Grab this app, and you will be bombarded with lots of weather info to suit your fancy. Great must have app. and its free.
*MoboPlayer - A nice app that plays most videos you throw at it. Especially those .flv and xvid videos you would love to have access to on the go!
*Trapster - Some people (Politicians and law enforcement officers) call this a DUI Checkpoint app. Others (me) call this A driver's social networking app. It basically uses user supplied data (and other means) to inform you of DUI Check points (also known as uninsured driver and driving without a license checkpoints), speed cameras, and popular police hiding spots. Just run the app while driving and it uses your GPS to determine if you are near one of these reported check points. For now, Google is keeping this one in the market, though Apple has decided to remove all such apps from their app stores.
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I also recommend these apps, which cannot be found on the app market.
HTC Flashlight: This app came with the Froyo upgrade on the original Evo 4G. Its a cool app that uses the flash LED light on the back of your phone as a flashlight. it has 3 brightness levels, a light flash mode and an SOS mode. Find the apk by searching the forums.
Amazon App store: head over to amazon.com and they will give you complete instructions for obtaining the app for your phone. Its a VERY simple process. Its basically an alternative app store. Some publishers release their apps exclusively to that store for a while. And they make a paid app free everyday for 24 hours.
Netflix: This app is not yet available in the market for the Evo 3D. But the apk works great. Grab it by searching the forums. It should not be hard to find.
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These are my favorite apps. I have more, but these I use regularly