Don't go alone. There are a great number of hazards hiding under the snow and the impact of being caught out alone is much higher. Go with a friend. Sorry, Treen, you don't count. Go with a human. An adult human that can effect a rescue if necessary.
Don't walk on snow bridges. You know, those weakening structures that are undercut by running water. If you can hear water don't step there! As the season continues the bridges will become less reliable and you're more likely to fall through.
Don't rely on your memory to follow the route. Snow makes everything look different. The trails are buried and the trees are shorter, bent over, or missing entirely. You might have walked this way many times before, but make sure you have extra navigational tools like a compass, map, and GPS.
For goodness sake, don't just follow someone else's footsteps! If you might make mistakes what makes you think the person that was here yesterday knows where they were going? Even worse is following a set of tracks is following old tracks that only go one way. (Why? Use your imagination.)
Don't get wet. Snow is cold. Somehow snowmelt is colder. Whether it's an exposed creek or a puddle, stay out of the water. Gore-tex boots will help, but not completely. Cold toes are not good when you're miles in the backcountry.
Finally, don't "go light" to go faster. After a long winter of carrying tons of gear you can't get complacent just because the snow is consolidated. A night out in near-freezing temperatures is almost as bad as a winter night out. And I'd say getting caught in a spring rain is worse than being in a winter snow. Carry your essentials and be sure you're ready for any eventuality.
Yes, we made all these mistakes. Treen and I were way ahead of everyone else so we were effectively alone. I punched through a snow bridge I thought was solid and wound up hanging over a head-high void. I took the wrong gully up and had to do a little extra scrambling. The footsteps I followed? Turns out they were a deer's. I stepped in a creek because I was too lazy to find a better crossing. And to round it out, I didn't have a full complement of emergency gear.
Nothing (too) bad happened on this trip, but it was a great reminder to make the best choices when heading out. Do learn from our mistakes and do get out there. Spring and early summer are wonderful times of year to be in the mountains as long as you keep your head about you.