If we picked just one or two shows to see, what should they be?
Since you're going during the summer, they have different night time shows during the Electric Ocean event. These night time shows are better than their regular daytime (year-round) counterparts so I definitely recommend them. The best of these shows are Sea Lions Tonight, Shamu's Celebration: Light Up The Night, and Ignite (pyrotechnic/fireworks finale that is superb). If you just stuck to the night time shows, then you could focus on the rides, exhibits and any animal encounters you wished to do during the day.
In terms of ride efficiency, should we head clockwise or counterclockwise, and is there anything we should hit first (or last)?
If you arrive at park opening, you'll have to go left. The park does staggered openings in which the right side of the park opens an hour later (so if the park opens at 9, the whole right side opens at 10). It's good to arrive at the park 30 minutes before it opens if you're able to as that's when the parking lot opens. It would give you time to park and walk to the gates with some time to spare if necessary. There are bathrooms available if needed also.
Once the park opens, I would suggest doing Manta first*, then head straight to Antarctica to ride Empire Of The Penguin. That ride will have one of the longest lines all day (at least until Electric Ocean kicks off later in the evening). I would then do Journey To Atlantis (as it's line can get long as well). Kraken is right beside there, but the lines generally aren't too bad for it (though it has been closed the last 2 days for maintenance...no word on when it's reopening). I would then head to Infinity Falls if you're wanting that as it's line usually is the longest when it's working. You can hold off on Mako until after the others since it's a people eater. *Alternately, you could start with Empire Of The Penguin, then Journey To Atlantis, Infinity Falls and then head back and do the big 3 (Mako, Kraken, Manta). Of those 3 however, Manta gets longer wait times simply due to it's loading design, but hasn't been too bad lately.
If you are not wanting to crisscross or back track too much (understandably since it's likely to be hot), the just start to the left with Manta and work your way around clockwise. There are many exhibits and aquariums to view along the way, especially the Dolphin Nursery, Manta Aquarium, Stingray Lagoon, Dolphin Cove, Turtle Trek (Sea Turtles & Manatees), Alligators, Sea Lions & Seals, and so on. You would come across Journey To Atlantis, Kraken & Empire Of The Penguin in that order (and that's just the left side of the park).
You didn't mention if you had children coming with you. Even if you don't, it's a must to head through Sesame Street. There is a lot to see if you're just looking. Of course if you have children, they'll love the rides too. There are character meet & greets with Elmo & Friends (usually Cookie Monster but have seen Abby and some others with Elmo inside that area), a parade that goes on 2 or 3 times a day, splash pad and various other activities. Some of the characters also walk around the land periodically and will pose for photos.
If you really like Goliath at SFOG (as I see in another post above), you're going to love Mako. I find Mako better in terms of airtime. Mako is currently my favorite coaster in FL. Manta is also MUCH BETTER than Superman: UF.
Are there any rides or shows that we should avoid?
Wild Arctic is really the only ride not worth a wait for. It's old and soon to be replaced. There is a no ride line however so you can see the belugas and walruses in their exhibits, so don't miss that.
As mentioned above, the shows are better at night. I do like the recently revamped Sea Lion High: The New Class. However Sea Lions Tonight is still much better. If I were to choose between them it would be Sea Lion High. The same goes for One Ocean (orca show). The night time show is much better and more high energy (Shamu's Celebration Light Up The Night). I do like Dolphin Days, but if you're pressed for time, you could skip it. Pet's Ahoy is great too, but it can be skipped if time doesn't allow.
What’s the best animal encounter?
If you're looking to do a paid encounter, the best would be Killer Whales Up Close. It's a little pricey but worth it from those I know who've done it, and pictures I've seen. If you want one that's not as pricey, then Dine With Orcas is a good one. You get a buffet style meal with a more intimate show with the orcas (& pilot whales). As far as the free exhibits, Dolphin Nursery, Manta Aquarium. Dolphin Cove (especially the underwater viewing), Penguins (part of Empire of the Penguin but does have a non-rider line for the exhibit only) & Shark Encounter are must do's. A tip about both dolphin areas (Nursery & Underwater viewing): Dolphins react to various objects like fidget spinners, keys, slinkys, even stuffed animals. You can put them right up to the glass and move around and they'll follow you and interact. We did that last time.
{EDIT} I see you may be arriving in the afternoon? I would still follow the clockwise pattern I mentioned above (to the left). Just do the rides as you come to them. If pressed for time, you could actually just skip the Empire Of The Penguin ride, and do the non rider line to see the Penguin exhibit. It's nice and cold in there and definitely worth a look to see them. I'll go ahead and still say skip Dolphin Days, Pets Ahoy, One Ocean and Sea Lion High and do Sea Lions Tonight, Light UP The Night and Ignite. Still check out Sesame Street though, it's not out of the way.