Re: What do I need?
You have several options depending on what you want to achieve. The bottom paint you see chipping is most likely an ablative paint used to keep barnacles and growth from attaching to the hull. Ablative paints are made to slough off over time keeping growth from getting hold. They normally only last about two seasons. Most saltwater boats that live in the water get bottom painted every two years or so. That's why you have the chipping. It is possible to simply give the surface a good cleaning and sanding and recoat it with bottom paint. There are ablative paints on the market that can be recoated even over an unknown bottom paint. After all it is meant to come off.
Recoating it with ablative paint is not the best option on a trailered boat. The paint will come off on the trailer especially if it is a bunk trailer. Roller trailers are a little more forgiving but the paint will not last the loading and unloading process. It will generally look bad in short order and you will likely get tired of painting it. But it is an option.
You could remove the paint using stripper as TheWoodCrafter suggested. In fact there are specialty strippers on the market made for the task. After stripping the paint it may be possible to restore the original gelcoat. It is very labor intensive but possible if the underlying gelcoat is in good enough shape. If fillers were used for previous repairs to the bottom, and the surface is not pristine, restoring the original gelcoat may be out of reach.
The most durable option would be to re-gelcoat the bottom after stripping. New gelcoat will definitely hold up best on the trailer. Applying gelcoat is far more complicated than paint however. It has its own set of difficult challenges. I have seen gelcoat applied with a brush, sanded, then buffed. It looked ok at best in my opinion. Gelcoat can be sprayed...I hear, but that requires specialized equipment and allot of know-how. It is generally not recommended for us D.I.Y types.
I would recommend repainting the bottom. Start by stripping the old paint off. If you leave the old paint, and decide to recoat with bottom paint, it will look like it has been recoated. No matter what you put back on, stripping the old paint will make it look better on the trailer. After stripping assess the bottom to see if the gelcoat can be restored. If fillers are present or if there is allot of damage to repair I would go with paint. One thing to consider is the trailer. Bunk trailers are rough on paint. I would use only a high quality marine paint. There are hard modified epoxy antifouling paints and self polishing bottom paints that are better for trailering. You could use a good two part topside paint and get a great finish. Be advised however that topside paints are just that, topside. They are not designed for emersion in water. Interlux says their two part topside can handle only about 72 hours under water. This time is reduced after it gets scuffed across a trailer bunk a few times. I am willing to risk it on my little trailer boat. I will be using two part on it. I am considering converting my trailer to roller bunks to help it hold up. I have heard from others in here that they have had good luck with it on their trailer boats.
I hope this helps you make a decision. I know how hard it can be.

Drewp