Get a sturdy case
Kids drop things - a lot. (Well, so do I, but that's not important.) A rubberized case will protect your tablet from most drops, so it's well worth the investment.
Tablets come on all shapes and sizes, and so do cases. The Speck iGuy case for the iPad shown above is one option, but if you need to look further afield, here are a few websites where you can buy a variety of cases:
Perform a factory reset
If you've been using your old tablet for any length of time, chances are good that there are any number of things on your tablet that you don't want other people to see.
It might be as simple as work emails you don't want anyone to read, or perhaps you don't want your reading habits shared with the world. In any case, the best way to make sure that all of your personal documents are removed is to reset the tablet to factory settings. With most tablets this will wipe all of the internal storage, remove any apps you've installed, and delete any changes you've made to the settings.
The steps involved vary between tablets, but you can probably find the reset option under the Settings menu on your tablet. Just make sure that you save any important files first.
Install a Parental Control App
If you would like to restrict what a kid can do with the tablet (block games, inappropriate content, and so on) then you should install a parental control app. I'm not sure I would take this step, but you might want to. There are a lot of websites, Youtube videos, and apps that aren't appropriate for small children.
There are a number of apps which would work. One Android app you could try is Kids Place. I came across it when I reviews the ClicknKids tablet, and on that tablet it blocked access to any app which had not been selected and approved by the parent. I thought it worked well, but if you don't like it there are many other alternatives in Google Play.
Of course if you have a Kindle Fire, that tablet has parental controls built in. So does the iPad.
Block in app purchases
We've all read the horror stories about kids running up thousands of dollars in charges, right? You don't want that to happen to you, so you should probably make sure that your tablet asks for a password before any purchases.
On Android tablets, both Google and Amazon have a parental control option under the settings menu which can be set to require a password before any purchases can be made. On the Kindle Fire, this setting is found with the other parental control options.
On the iPad, I would suggest a different approach. I would set up a separate account just for the kid's iPad. This is extra work, but it will allow you to remove any chance that the kid might run up charges in iTunes.
If you need to make purchases for the kid's iPad, you can buy a gift card with the parent account and give it to the child account.
Install a Kid Friendly Web Browser
As a final step, I'm going to suggest that you install an alternate web browser. There are any number of child safe web browsers in Google Play and iTunes.
The 5 suggestions in the above post are really just a start on what can be done to child-proof a tablet. If you have any suggestions for other apps to install, don't hesitate to leave a comment.