The answers to your questions depend entirely on where you live. A federal law passed in 2006 (UIGEA) made it illegal for U.S. banks to process payments to or from online casinos. This makes it difficult for U.S. citizens to make deposits into or withdraw funds from online casinos. There are now legal exceptions to that law, discussed below.
Let's start with the easy answers, which assume you live outside the U.S. You can google "online casino watchdog sites" and you'll quickly land on the reputable ones, where you'll find honest reviews of online casinos from players who report their experiences. If you avoid casinos that either aren't reviewed on these sites or where players on these sites report problems, you'll be playing in "safe" casinos, where the software is likely honest and the casinos pay promptly. You'll also find full info on how to download the gaming software and deposit and withdraw funds, etc. on these watchdog sites, as well as on the specific online casino sites.
If you live in the U.S., it gets a bit more complicated.
If you live in New Jersey, you'll find a list of seven online casinos where it's legal and safe for New Jersey residents to play casino games. Depositing and withdrawing funds from these casinos is as easy as making a purchase on Amazon.
If you live in Nevada, the only fully legal online gambling option available is for playing poker the World Series of Poker site.
If you live outside New Jersey and you want to play casino games or slots, you're in more of a legal gray area. Most experts believe that it's de facto legal for U.S. citizens to play in online casinos. I'm unaware of any players ever being charged with a crime for playing online and the only penalties ever levied against such play that I know of are levied against banks for processing the payments.
Many online casinos state in their terms that they don't accept U.S. players, but some do accept them. The ones that say they accept U.S. players usually accept credit-card deposits, wire transfers, or Western Union fund transfers. They accomplish this by setting up a third-party account (like Joe's Sausages) and they pay players who withdraw funds via checks or wire transfers from these third-party accounts that have been set up to fool the banks. The problem you'll face is that banks sometimes discover where these third-party checks are coming from and if your bank deposit is rejected, you'll have to notify the casino that they'll need to try another third-party account to send you your withdrawal.
The best way to assure yourself that the online games are honest is to play only in casinos that use software that has been deemed honest. MicroGaming software is honest, as is RTG, Playtech, and WGT (and many others). If you stick to gambling sites recommended on Casinomeister.com and Askgamblers.com, you'll probably be playing against casinos using honest software.
Casinos that run on RTG software usually state that they accept U.S. players. In most cases, U.S. players can't click through from Casinomeister.com or Askgamblers.com to an online casino link, even if the site says that the casino accepts U.S. players. To get to the casino you'll usually have to input the url yourself.
Some U.S. players avoid bank transactions entirely by depositing and withdrawing via bitcoin. At online casinos that allow you to make bitcoin deposits, you can also only withdraw your funds in bitcoin. You'll have to convert bitcoin to back dollars through independent bitcoin exchanges.
If you're not familiar with bitcoin, start by reading about it on Wikipedia to get a minimal handle on the methods of buying and selling bitcoin and the types of problems associated with using this open-source payment network, which can be a gamble in itself.
An increasing number of honest online casinos (at least a hundred of them) now accept bitcoin gamblers.
But before depositing funds in any currency at any online casino, check the reviews at Casinomeister.com and Askgamblers.com. If an online casino is not reviewed and recommended by one of these sites, steer clear. A lot of crooks are waiting to pounce on you out there.