This is my opinion, based on years of interviewing and hiring.
Don't bring up the request for benefits materials at your first interview, maybe they will provide it? The first interview (and who knows, it may be your only interview, some places interview once and make offers, others may have a process where you have to return for several interviews...) but during the initial meeting, you need to sell yourself. Make them WANT to hire you, that they can't NOT want you.... then when they make the offer, ask for a copy of the benefits handbook or materials, you could also ask for an employee handbook but they may not provide that depending on how they utilize it. It's not unusual for you to have to "sign" something to receive an employee handbook, or as is the case more and more often, the handbook could be "online" and you have to be set up with a username and password to access.
But when they make the offer, ask questions about the benefits and ask for whatever materials or information they can provide you about the benefits. Remember, your benefits are part of the overall compensation so it is important and understandable for you to ask questions and get information regarding them.
Now if they ask you to start right away, personally, I would always say that I really feel it appropriate to provide my current employer with the customary 2 weeks notice, but I would certainly be willing to discuss it with my current employer as I haven't been there long enough to know what is customary there yet. To be honest, I have made many offers asking people to start a.s.a.p. and I sometimes lose a bit of respect for those who are too quick to agree on giving less than 2 weeks notice.
Most employers, even those who are very eager to have you start right away, will understand and respect your need to provide notice. Maybe you can work out a compromise where you work part time at both jobs for a week of transition. I've seen that done as well.
On the otherhand, unless you signed a contract at your current employer, you are most likely in at "at-will" employment situation and are under no legal obligation to provide notice. BUT you might want the reference later, or at least not to ruin a possible decent reference.