Copying your PSP save data
Plug your PSP into your PC with a Mini USB cable (the kind that fits into the top of your PSP) and select the USB Connection option in the PSP System menu.
The PSP should show up as a connected drive in your My Computer menu. It's probably called "Removable Disk" if you're on Windows. Open that drive and navigate to the PSP folder.
Copy the SAVEDATA folder onto your PC somewhere like you Desktop. That folder has all your PSP saves and we'll copy them to the PS Vita soon. If you're concerned about how much space the data will take up on your PS Vita, check how large this SAVEDATA folder is, you're about to copy that much data onto your Vita. If it's too large, we can only copy the specific games you want the data for later.
Prepare your PS Vita to copy the files
Download any one PSP game onto your Vita and play it enough to generate a save file. Any game will do, it doesn't have to be a game you want to copy the save data for.
Install the PS Vita Content Manager Assistant for Windows or Mac OSX.
Connect your Vita to your PC with the USB cable and open up the content manager. Pick Copy Content: PS Vita System -> PC -> Applications -> Save Data (PSP/Other).
You should see the game you played earlier in a list; tap the PSP game to select it and hit copy. Click Okay to copy it onto your PC.
Find the folder on your PC the Vita copied to; it should be in your Documents folder in a folder named PS Vita, in that folder there's a folder called PSAVEDATA. In that should be a single folder named with random letters and numbers. Open that folder; this is where we're going to copy the PSP Save Data.
Copy the files onto your Vita
Open up the SAVEDATA folder from earlier, those are your PSP saves. Also open the PS Vita folder we mentioned before that has your PS Vita's copy of the save files (you want to be inside the folder with the name that's random letters/numbers). Copy and past ALL folders from the SAVEDATA folder into the folder named with random letters. Say Yes to any dialogs that ask if you want to merge/copy files; this means you're overwriting the Vita's copy of the save file with the PSP one.
Now just open Content Manager on the Vita again and pick Copy Content: PC -> PS Vita System. Pick Applications -> Saved Data again. You should now see all of your PSP save data. I recommend hitting Select All and just copying all of this data over. You can also pick and choose just the games' data you want to copy over if you want to save yourself some space.
Pick Copy and hit Okay again, it will ask you if you want to overwrite any data you have on the PS Vita for the PSP games; you want to do this, so click Apply to All and tap Overwrite.
Your PS Vita now has all save data from your PSP and you can keep it there forever, and it's also stored with your PS Vita backup. Any time you download a PSP game onto that Vita, it will start with the save data you copied from your old PSP, no need to repeat this for each game!
Best Answer
UPDATE: As of Vita firmware Ver. 2.00, the browser is now a "tiny app", meaning it can be run alongside game software without quitting (or having to do the Twitter workaround).
Original answer follows:
While the Web Browser isn't directly accessible while running a game (Vita, PSP or PSOne), there is a workaround to surf to websites while using software.
What you need to do is install the LiveTweet app from the PSN store (available for free). This app is accessible while running games. Either enter your twitter credentials and look for a tweet with a link or use the link to authorize the app that's available if no account is entered, clicking the link will open the web browser w/o closing your currently running game and you can browse to whatever guide or eBook you need, please note that minimizing the browser (using the PS button) will require you to enter it again via the Twitter app. Open webpages will need to be reloaded, so make sure you have internet access when you use this method.
Here's a nice video guide on this workaround:
Another semi-awkward method to use guides while playing games is saving the guides as images and loading them to the picture app, this is actually a bit more convenient if you need to move quickly between the game and guide because you can do so without using other software in the middle, but you'll lose the convenience of searching through the text. You can either use the Vita browser and take screenshots (PS+Start) or use a Chrome extension like Webpage Screenshot and transfer the files over to your Vita.