Florida to Yellowstone early April

I've been in Yellowstone mid May and had snow quite often that week. There were three mornings that the park roads were closed until mid day. The park itself was nearly empty which was great compared to summer. Had trout streams all to myself and caught trout on a Dusty Miller fly while big heavy snow flakes filled the air around me. Hardly any cars. Had hiking trails to our selves. The only park facilities open were at the four main junctions. Snow was pretty deep on many of the trails but we had a great time. Been there other times when it was wall to wall people and not as much fun, but still great.

Agree with others here on early season Yellowstone weather.

I've been there the first week of June and had snow flurries hit.

For the most part, you will be good travel weather, but as said previously, once you start into the higher elevations, almost anything can and probably will, happen.

April is the 2nd snowiest month here in Colorado. For all areas of the state, not just the mountains, March being the first. Storms can be frequent in early April, and usually heavy and wet snow, versus the more powdery snow earlier in the snow season. All of the interstates can be a challenge during a storm and immediately afterward. Eastern Plains/Colorado Outback blizzards can be common, with whiteout conditions and high winds.

Timing is everything. If you're considering, for example, coming in on I-70 from Kansas and then heading north on I-25 in to Wyoming, and are prepared to hole up somewhere in the event of a storm and wait it out, then that can be a doable route. I-25 in Wyoming can be a real bear to drive, due to the prevailing wind direction (you'll be in a goodly head wind most of the time). Again, timing is everything.

As stated, April is a very iffy month for travel. you're liable to see sleet or snow when you get close to the continental divide if you're north of I 10.
Or
You might just sail right through without a problem.

But Yellowstone doesn't open its roads until mid May as a rule.

I-40 was closed just west of Albuquerque for two days at the end of April one year for us. You can't count on weather. Just allow yourself more days than you think in case you have problems. April still brings snowstorms and ice. Be careful and monitor the weather as you travel. Good luck!

Early April is always going to have some small risk for snow/ice anywhere north of the deep south especially getting into higher elevations. It sounds like you don't have a choice of starting date so just watch the weather reports, be prepared to stop and delay and stay south as much as possible.