I have installed engineered hardwood floors in my living room and hallway. I did not extend the wood into the closets, I just left the carpet inside those since the carpet will not be seen with the doors closed. What is the best way to hold the carpet down and cover the ends of the wood boards?

Every flooring company makes a variety of transition pieces. They make a trans stick especially for wood to carpet. Look in the laminate flooring area of a home store.

When you install the transition piece (there are a couple of different types) just be sure the carpet is secured to the floor . If you use a piece of flat metal (would not be my first choice), you can use a piece of standard oak trim you may have to play around with it to make it fit just right. This would be my second choice. My first choice would be a transition used for laminate a plastic track goes down then the moulding just pops in. If you use this kind simply cut the carpet back a little about (7/8 of an inch if I remember right) remove the carpet and pad there should now be a gap between your carpet and your wood flooring. Now put the track down and screw it into the subfloor don't overtighten, it can warp the track. Next use either a narrow crown stapler or some small nails and tack the carpet to the floor. Finally cut your transition piece to fit and snap it into place. If I have missed anything or if you have any questions feel free to post back.

Whatever floor you have will have a few different types to choose from. The one it sounds you need would be a t strip. If your looking at the end of it it's a t shape, you'll need a 1/2 gap between the carpet and floor to install the black channel. Its plastic and you can but it down with sheetrock screws 1 1/4 is fine to use. After this cut the t strip to length and just push it into the channel, there are teeth on the t strip that correspond to the channel and it will grip it tight, make sure this is the right size because you only have one shot at this, so if it's a 1 1/16 short its OK just not more than that,
The strip itself will lap over the floor and the carpet,