The kit comes with a 7.8 gallon fermenting bucket and a 6 gallon glass carboy. Just so people are aware.
Firstly, it's up to YOU to decide if you want to use a secondary or not. This will likely determine how you use this setup. I highly suggest you read up the MANY MANY threads already debating the pros/cons of a secondary. Suffice to say that there is no perfect answer for everyone. It's a matter of convenience, mostly. The purpose of a secondary is to get the beer off the yeast to prevent the dead yeast cells from autolysing and creating bad flavors. Many people had fermented in a primary for many weeks without issue. Some say they can taste the difference.
For using a secondary: Ferment in the bucket. After the primary fermentation rack to the carboy. After the secondary, use the bucket again as a bottling bucket.
For NOT using a secondary: Ferment in the carboy. After primary fermentation and maybe 1-2 weeks of secondary time, rack into the bucket for bottling.
How long you do the primary and secondary ferments is going to vary. I cannot tell you exactly how long is long enough. You will have to use the hydrometer and take readings. After the beer has reached expected FG (Or very close and stable over several days) you can rack to secondary. If you are not using a secondary then just give a week or two for the beer to clear up and then bottle.
I personally keg, so when it's time for me to secondary I usually just rack to the keg and let it sit for a week or two before moving to the kegerator. IMO there is nothing wrong with using a secondary. When you think about it, the only downside is that you have to rack one more time, which some consider to be a risk of infection. I don't think it's a real concern. There are many times you expose the beer to the air for a few minutes. Unless you are extremely careless, there is little chance of picking up a real infection. The upside is that when you rack to a secondary you can use the primary again for another batch. In your case, though, you don't have a dedicated bottling bucket, so you are kind of SOL on that regard.
If I were in your shoes, I'd buy a dedicated bottling bucket and use your current bucket for a fermenting other batches. Fermenting buckets are pretty cheap.