I found a cat/ankle biter dog sized hard carrier at either Ollie's or Big Lots for less than $20, then got one of those cat beds for $4 for inside. If you have some type of one of those outlet stores in your area, check it out if you wish to save a few $$$
I was at a chain pet store after that purchase and they had the same size for like $30. A bit better quality but the one I got is secure, safe and built well. LB is hardly ever in it so I didn't see the need for top quality.
I highly recommend the sort of transport carrier that people use for show rabbits. They can be purchased from online rabbit vendors. The wire floor keeps them clean during the trip and gives them more traction than a plastic floor. The wire sides allow for more airflow. And because they open from the top (without any sort of lip for the rabbit to hold on to) they are 1000x easier to get the rabbit out of than any other type of carrier. People that show rabbits transport rabbits all the time to and from shows, so they generally know what's best for transport.
They also sell it with multiple compartments if you have multiple rabbits to transport. Also in different sizes; most rabbits fit in the standard size but there are larger carriers for giant breeds. If you rabbit is easily stressed by carriers, you can fill it up with hay to cover the sides a bit and give them something to munch on during the trip.