What's the best way for a DIY person
to paint the interior of a new house?

Roller is good if you're a pro and know how to do it without getting a lot
of lines.

New house is infinitely different than doing it with stuff in there. I used
a Graco XR7, and was pleased as punch. You have one person spray, and
another roll right behind them with a sheepskin roller. Have several
rollers, and change to a drier one as that one loads up. You get lots of
paint on there for a first coat, and the backrolling puts a nice stipple on
it. Once a roller gets full, just wash it with a water jet, no need to take
out all the paint to use it for backrolling. In fact, don't use one dry,
but put some amount of paint on it before you backroll, or your stipple
pattern will be noticeably uneven.

You also get overspray on EVERYTHING, so prep is vital.

Also, spray the walls first if they are going to be a different color than
the ceiling. Let the spray go up enough to get into the corners, a little
onto the ceiling. Let dry for a couple of days. Tape off the walls with
GOOD TAPE AND 18" strips of paper. Now spray the ceiling, backrolling it,
too. Pull the tape and paper with the paint wet, or you will pull off a lot
of stuff you don't want to. Do the final cut in with a one inch tapered
brush.

I know that paint sprayers aren't the most popular ways to paint, and there
are lots of nuances to them, but if you are doing NEW construction, and the
overspray won't get on furniture or appliances, it's the best way IMHO to go
and get a thick coat of paint on evenly and have it look good once it dries.

The sprayer also shines when spraying over popcorn. Mask and spray, and it
looks like new. Although, it is about as easy to just remove the popcorn,
and once you've painted the popcorn, it is a pure D bitch to get off
compared to unpainted.

No comparison outside for block walls and open areas as well.

Don't waste your money on the wanna be Wagner Power Sprayers and the like.
I got an XR7 on ebay for $325, and it has saved me that much, and more.
Plus, two guys can paint 250' of block wall before lunch. Try that with a
roller.

Just my experiences. YMMV.

Steve