I've been home brewing for several years now. I started with extract brewing, then went all-grain. I've used a number of methods for carbonation during that time.
Table sugar added into the bottle before filling and capping: This is relatively inexpensive and easy, but can be a little messy. Some say you get off-flavors from this, but I've never noticed it. It requires you to figure out the right amount of sugar for the amount of beer you're bottling and the carbonation level you want, but it's a decent solution in my experience.
Corn sugar dissolved in boiling water, cooled, and added to the bottling bucket: This is probably the most common method used, and possibly the least expensive. It yields the best control over the amount of carbonation since you calculate and measure out the sugar based on the desired carbonation level and batch size. It's not as easy as the table sugar method, but not exponentially more difficult. Most home brewers swear by it. I've gotten good results from it as long as the wort whirlpools around when I transfer it to the bottling bucket (and thus mixes the sugar in well). In place of corn sugar, I've also used candi syrup successfully for this.
Carbonation drops like these: You add one of these to a small bottle or two to a bomber sized bottle. No boiling, no measuring, and you know you're getting a consistent amount of sugar per bottle. The down-side is that you can't control the precise level of carbonation as you can with the other two methods above. You get the amount of carbonation the drop provides - no more, no less. This can be too much for some styles and not enough for others (like the Belgian Tripel, for example).
In my experience, all three methods (and their variations like using maple syrup in place of corn sugar) all work. Executed correctly, they provide the sugar you need for the yeast to carbonate your brew. If I'm short on time, I will use the carbonation drops because they're quick and easy. Tear open the package, drop one or two in a bottle, fill it, cap it, and you're done. If I want a precise carbonation level, I'll calculate the amount of corn sugar (or other sugar) I need, boil it up, cool it down, and add it to the bucket. That will get me a style-perfect level of carbonation. If I'm out of corn sugar and carbonation drops, I'll use a special measuring spoon I have and use table sugar.
I've never experienced any off-flavors as a result of using any of the above methods.
The drops in my experience will fully dissolve in the bottle and leave behind no residue. They provide an average level of carbonation provided your yeast is healthy and the bottle caps seal properly. For some styles, like the Tripel, I'll sometimes add an extra drop per bottle. You just have to be careful in these situations to make sure you use thick bottles. (The kind that Gulden Draak, Trappistes Rochefort 10, and other Belgian beers are packaged in have proven strong enough in my experience.)
I've used these Coopers drops, Mangrove Jack drops, Brewer's Friend drops, and others. All of them seems to be equally good. The Brewer's Friend drops seemed to stick together inside the container really badly, which made using them a pain in the backside and resulted in irregular shapes (and probably uneven carbonation). The Coopers and Mangrove Jack drops didn't stick together and seemed to work well.
Here's a "pro tip" for those new to home brewing. Always assume that you're going to have a bottle which leaks or explodes. (You probably won't, but making this assumption will save you from unpleasantness later.) When you finish bottling a batch of beer, put the bottles inside a sealed beverage cooler or plastic bin with a locking lid. Let them condition inside this container until they're ready to drink. If something goes wrong and one or more bottles explode or leak on you, the mess will be contained within the cooler or bin. That's a lot better than having glass and sticky beer sprayed across a room in your home or apartment. (So far in 25+ batches I've not had a beer explode, but I did have a bottle of apple wine explode into shards. Fortunately it was a mess inside the plastic bin but that was a relatively easy clean-up.)