If you want to record vocals against a click or any tracks you've already recorded you'll need closed back headphones to monitor with. Open backed headphones allow sound to leak out, which microphones can (and usually will) pick up. I'd suggest adding Sennheiser to the list of phones to consider, the HD25 range is pretty good. They're aimed at DJs who need to hear one thing while a PA is pumping out something else, so have good isolation (low leakage) and are fairly honest, at least as honest as most affordable headphones get.
Headphones aren't ideal for mixing, because they present a very different picture to speakers, especially regarding panning and the problem where a sound in one stereo channel negates something in the other. Usually obvious on speakers, but not headphones as the sound from them doesn't blend in the air before it reaches your ears.
A built-in sound card might be OK if you just use it for playback, but they are usually very poor if used for recording purposes. Their microphone input preamps are generally very noisy and low quality. If you use software synths, samplers etc. or need to monitor anything through the DAW software built-in sound cards are unlikely to provide low enough latency to be usable. Latency being the time it takes for the audio to get into the computer, be processed and sent out again or the time it takes for a software instrument to produce the sound and it then to be made audible. This is easy to spot because you'll get a noticable delay between triggering a note and hearing it.
Avoid ASIO4ALL if possible. Some people on some forums seem to recommend it as an "ideal solution", even suggesting it be used when the interface manufacturers supply a very good driver of their own. It isn't ideal, far from it. It's an admittedly clever and complicated hack that attempts to compensate for the lack of an actual low-latency ASIO driver, but it's still a complicated hack. It can be very difficult to configure and cause havoc in all kinds of ways, including interfering with other audio drivers on the system.
There's a lot to consider when putting together a DAW and home studio, but that's the nature of the beast I'm afraid.