How can I get mods on a non-bukkit Minecraft server?

It really depends. Are you looking for a server with mods, such as Tinker's Construct or something along the lines of that, or are you looking for a server with plugins, like WorldEdit, WorldGuard, Essentials, so on. If you are going with the first option, your best bet will probably be using Forge, or some modpack that you can find on the TechnicLauncher, ATLauncher, both having easy server set up, assuming you know how to start up a basic vanilla server. The second option, Bukkit, may not be the one you are looking for. Bukkit does not work with mods. However, there are plugins out there to customize your Bukkit server, but those are more for server functionality, not like new blocks or anything. Bukkit still revolves around Vanilla Minecraft.

Your next question about copying your world is also dependant on which option you choose. You can certainly easily copy your world over to Bukkit, as it is still Vanilla-based. Just copy the contents of your world into the world folder with your server jar, and you should be fine. With the modded server, it depends on what mods you use. If you use something that involves terrain/world generation, you may ruin your world with that.

Looking back at your question, I think that you mean mods, so copying your world over may not be the easiest task...

Whether you'll need to switch server depends on what mods you want to use. Mods are developed for certain "delivery means" - either they plug in to Bukkit, Forge, or other mod manager, or they are supposed to be installed manually.

Manual installation means that you open minecraft_server.jar as a ZIP archive, take mod's files and replace similar files inside the game. This is least convenient, and there aren't much mods designed for that anymore.

Forge is a "plugin interface" for vanilla game. It means it is a mod that is installed manually (or via a "patcher" progrem that essentially does that for you), and whose responsibility is to "plug in" other mods into the game. With Forge installed, adding Forge mods is as easy as dropping mod's archive into /mods folder and restarting a server.

Bukkit, Spigot and others are custom servers, built from the ground up by people separate from Mojang. They are, of course, supposed to run the game exactly as vanilla game does, so official clients can connect and play normally. There are small changes, of course, like dropped items behaving differently, but world format is the same, so worlds can be transferred between them (well, at least from vanilla to custom :-). Those servers have plugin interfaces built into them, so, too, just drop the plugin into a folder and restart.

Whether you will lose your world also depends on mods you choose, but in a different way. Some mods just change existing rules of the game or add monsters or items (like minimaps or airships) - those are fine. Mods that you will need to watch out for are mods that change world generation. If you play with default world generator, and then swap it for another, the game will run okay, but as soon as you reach the edge of explored land, the landscape will be torn apart and a whole new world will start beyond - the game has no way to "stitch" together old and new landscape.

On the other hand, this may not be the case. If a mod just adds new ores, without changing how the landmass is generated, there won't be any problems, you will just have to travel to new places to find those ores (because explored chunks are already generated and saved, no new ores there).

Since you're looking for mods but only want to use the vanilla server, your best option is command block modules/contraptions. The way these work is by putting a (very long) command into a command block and powering the command block, which automatically generates a command block structure that changes something in the world.

The way to get these things and get started can be very complicated or very easy depending on where you look. The most popular methods is(are) Gamemode 4 (YouTube playlist) and just searching YouTube for command block contraption (there are plenty of those but aren't optimized for lag like GM4 is). I recommend the first since the modules were made with all the others in mind and they all use a single custom crafter.

And with your switching worlds question, the GM4 will work as long as it is 1.8+ server (as you have stated).