At some point, you're going to draft your roving too thin and it's going to break. That may or may not happen before you come to the end of your first strip of roving and want to keep spinning with a new piece. Either way, the solution is the same.
Split the end of the roving in half and do the same to the unspun end of your yarn. Put the ends together, tangling them a little to help them "grab," and keep spinning.
Troubleshooting
If your yarn is tightly kinked and feels hard to the touch, you're not drafting fast enough and your yarn contains too much twist. You may not be able to keep up with your spindle when you're first learning--that's fine. You can let your spindle spin without drafting for a few seconds, then stop it (set it down on a table, don't just let it hang or it'll start to spin backwards) and use the pent-up twist as you draft out the fiber. Remember, drafting is just the process of adjusting how thick your roving is and then letting it be twisted into yarn.
If your yarn tends to drift apart at the slightest pressure--and if your newly spun yarn is always breaking and sending your spindle crashing to the floor--you aren't putting enough twist into the fiber. You might not be twisting the spindle often enough, or you might have accidentally twisted it the wrong way (remember, always to the right).
If your yarn is lumpy--and it will be--with thin sections followed by lumps of thicker sections (called slubs), rejoice. You're spinning novelty yarn. With practice you'll learn how to draft more efficiently so that you won't get slubs, but that means no more novelty yarn. Believe it or not, eventually you'll have to learn a special technique to spin slubs--and you're doing it the easy way now! Save your first yarn, by the way. It'll make you feel so much better in a few weeks to see how you've improved at spinning, and that slubby, uneven yarn really will make an interesting texture when knit.