I can't comment on Bryce as I've never been there, but I have done the trip from Las Vegas to Zion a few times.
Public transport/etc to get there is basically non-existant. It's possibly you might be able to pick up a "tour" from Vegas, but this will only end up costing more and taking longer - especially with two of you. Rental cars are generally cheap in Vegas (cheapest I've paid is
Personally I wouldn't recommend trying to do Zion in a single day. It's at least a 3 hour drive one-way to even get to the entrance of the park. Within the park (for most of the year, including the next 6+ months) there are (basically) no cars allowed, so you have to use the shuttle buses to get around. A single loop of the shuttle, without getting off at any of the stops, takes around 90 minutes.
So allowing some time to get into and out of the park, return driving, and a single shuttle loop you've already taken up over 8 hours! If that's all you intend to do then it's certainly possibly to do it in one day, but IMHO that would defeat the entire reason for going to Zion - the see the park, especially via some of its brilliant hikes.
Spending a night, either in Springdale (located within walking/shuttle distance from the park itself) or one of the nearby towns (eg, Hurricane, St George) will give you much more time to enjoy the park, and make the entire trip far more relaxing. Personally I've always stayed in Hurricane - about 20 miles away from the park, but with cheaper and easier to book accommodation than Springdale.
Zion is an amazing place - trying to do it as a day trip just isn't going to do it justice!