I have a new Ultegra triple. When I first got it, It wouldn't shift to the small ring. A couple of times when I kept pushing the brifter lever, I threw the chain onto the BB shell somehow. I adjusted the cable tension using the barrel adjuster, and now, this is my situation. If the cable is loose enough to permit a decent shift to the the small ring, I have chain rub on the outer plate when I'm in the large ring and the smaller cogs. If I tighten the cable to eliminate the chain rub in the large ring, I cannot shift into small ring. I know that you should not have to adjust the limit screws to eliminate chain rub, but it seems logical to adjust the limit screw for the large chain ring. Am I wrong? It is a new bike BTW and I'm too lazy to bring it in for an adjustment. Thanks.

You need to tighten one limiting screw so that the chain doesn't hop off the small ring onto the bottom bracket.
You need to loosen the other limiting screw so that the chain doesn't rub on the outer plate of the front derailleur when you are on the large ring.

Then, when the limiting screws are right, work on the chain tension. It should be firm when on the large ring, and not too slack when you are on the small ring.

Don't let it frustrate you too much if you can't get it. Take it to the bike shop. They should adjust it for free.

Are you sure the derailer is mounted correctly? In-line with the chainrings, and rotationally around the seat tube? The fact that your low gear both rubs AND drops the chain tells me there's more than just cable tension and limit screw issues going on.

Personally, whenever I encounter a basket-case derailer, I start from the beginning and treat the problem as though I were installing the derailer from the beginning- unhook the cable, turn the barrel adjuster in about 75%, realign the derailer, check the low limit adjustment, connect the cable, then check the high limit adjustment and cable tension, in that order.

The front deraileur should be parallel to the outer chain ring with less than 2 millimeters of clearance to the outer ring. Cable tension adjustment should be that the rear deraileur is in the high gear cog in the back(smallest cog)and the chain just barely clears the outer cage plate of the front deraileur in the middle sprocket, front. This should be enough cable tention to get the big ring unless you set the limit to tight. You should also be able to shift into the inner ring as well from the middle ring in high gear as well. If not then adjust FD till you get this set and then come back to rear derail and shift to low gear in back and recheck to see if you can tighten up the low gear adjustment of the front deraileur any further. This should put you back to square one and cable tention on a new bike maybe needs to be readjusted several times until you have about 3-500 miles on the bike. Then the cables can be readjusted and should stay that way for a long time. Good luck

1 . since it is a new bike yes take it in and have the shop look at it. For most shops the first tuneup is free ,it the breaking in period and cables do need adjusting.
2. to do this yourself, you need to shift to the smallest gear (granny gear ) in the front and the largest in the back (low gear) now adjust the low (L) screw until the you eliminate the chain rub on the backside of the cage. now shift to the small gear (high) in the back and the large one in the front and adjust the high (H) screw to eliminate the rub on the front side of the cage. now take the bike out for a ride and do any minor adjustment with the cable adjusters.

Thank you for your thoughtful reply! I feel confident in the mounting of the derailleur. I agree that my shop would adjust it at no charge, I'm just a lazy bastard and I feel it just needs a "Tweaking". Would you happen to know which limit screw is which? Thank you.

Most reliable and relevant method of figuring it out, in your case, would be to drop it into the low gear, slack the cable, and turn a screw clockwise- be precise, and keep track of how far you've turned it. If you tweak the correct one, the derailer will move toward the small chainring.

(Jumping between different brands and mtb/road makes remembering which is which difficult...)

Welcome to the wild wacky world of triple front derailleurs! Get yourself a jump stop so you will never again experience a suicide shift to your bottom bracket. https://www.gvtc.com/~ngear/whatis.html

The L screw is usually near the frame , (but not always ,sometime they are switch) and The H screw is the farest from the frame.

yup somewhat common to see this on dura ace or ultegra triples. to get a good shift to the small ring the inner plate doesn't guide the chain to the ring. i some times angle the tail of the fd outward a little to see if it helps from dropping the chain.