I use an anchor release technique.
In a nutshell, I have a ring spliced on the end of my rode, I have a ring on my rode that gets a buoy on it, and a short tag line on the ring on the end.
I run a line from my bow cleat, down through my bow eye (do this at the dock!!!) and then back to my transom cleat. The anchor always has the buoy in case of a whoops, but basically you just put the spliced end ring through the line that is connected from your bow and cleat it down to your stern, and have that tag line clipped to the ring that you keep the other end accessible from the cockpit. The anchor will quickly head up to the bow and hold you normally.
If you want to get off your anchor and come back you just uncleat at the stern, and let the ring off. It's good to make sure this line is not long enough to get into your prop, but even better to get it out of the water after you are off your anchor. Then you are free to fight a fish, rescue a man overboard, or do whatever, and can just come back to your anchor without hauling it up and setting it again. It'll just have the end of the rode floating up with the buoy.
When its time to pull the anchor, just get the buoy, then the rode, and haul it up in your puller or hauler like a crab trap.
When I get to the chain I just haul it by hand.
It should be said that I am in a prime position to use the boat to pull the anchor by cleating that line to my stern instead of putting it through my puller. The rode will come up to the surface, feeding through the ring with the buoy, including the chain, and when the anchor gets to the buoy it will conspicuously pull the buoy under. Then you stop and pull the whole thing in easily by hand with the anchor and chain suspended by the buoy at the surface.
I just don't love the idea, and was never brought up pulling an anchor like this on a boat. I have no problem using my hauler instead.
Youtube has a bunch of videos that describe this type of anchoring. I hope I explained it a little bit.
John