I use a couple of techniques for a high sheen.  Will it be used for food and washed or more of a decorative piece?

The easiest but not as durable method, especially if washed, is a friction finish.  You can make your own using shellac and BLO or tung oil but the Hut CrystalCoat or Mylands High Build Friction Polish are my go to.  The Mylands is a little cheaper of the two or at least you get more for about the same price, IIRC.  Its bottle is not convenient for application so I use a small condiment bottle to apply.  You sand your piece to at least 400 grit but I usually go to 800 or even 1000 sometimes.  You simply wipe on enough of the finish to thoroughly wet the surface and then use a small piece of cotton cloth to buff it out.  You turn the lathe speed up enough that the cloth generates heat and it quickly sets the finish.  I usually apply 2 or 3 coats.  You can apply several coats within about 5 or 10 minutes and it cures overnight.  You can carefully wash it but I would not submerse it to clean it and if washed often, like a salad bowl, you will have to refinish it after a while. 

A more durable option is sand to at least 400 grit and apply a few coats of polyurethane or spray lacquer.  You lightly sand between coats to knock down any nibs.  After the final coat is cured, you can polish it out on the lathe with a sanding paste such as Acks or Yorkshire Grit followed by a wax.  I really like the Acks.  It comes with the finishing wax.  You can use the sanding paste without applying a clear coat first but it will not be as high of a sheen.   Note that I sometimes use Minwax's poly sanding sealer by itself as the clear coat.  It is water based so does not change the color of the wood very much.  It takes more coats to build but it dries quickly.  

A third method is using the Beall Buffing System.  I have never tried it as the friction finish appears to be so much easier (and cheaper) and yields similar results.  

In both of the first two methods, sanding to a silky smooth surface is key.    You want the wood to have  a slight sheen after sanding even before you start applying finish.   I usually wipe the surface with alcohol or mineral spirits before and after the last grit to get any dust off before applying a finish.  Applying before the last grit may slightly raise the grain and the last grit will give you a smoother surface. 

--Nathan, TX. Hire the lazy man. He may not do as much work but that's because he will find a better way.