What would you add?

A glitch free gaming experience
A reduced animation for egg hatching, sending and opening gifts, and catching.
3 more slots for pre made partys
A slight reduction in the massive amount of catering to the casual, collectors, shiny hunters crowd.

For the Friends list, an indicator if you have earned a point with a friend that day. Would prevent you from opening a gift in the evening if the friend opened one earlier. I refuse to make a daily spreadsheet for this.

A hybrid of the old and new gym system: 10 slots, no duplicates, daily coin claiming, no motivation decay.

A return of some sort of training that can actually do something besides work towards a medal. Maybe add battles with (certain?) wild encounters that can increase your Pokemon's level.

Stop using RNG for *everything*. If a person can land 13 consecutive excellent curve balls with Golden Raspberries on a legendary, that person deserves to catch that Pokemon. End of story.

A new, more interactive combat system would be huge. More streamlined trading, friends system, and a another gym rework or more interactive gyms.

A good official alternative to scanners.

A new battle system. Tap and swipe is getting tedious. Level cap increase. I'm an F2P casual and even I am level 40. Plus I want to max my best pokemon above level 40. No more ridiculous bugs like this berry bug. Shame on Niantic for having such major bugs despite billions of dollars flowing their way. More challenging level 3 raids. With all these new pokemon like kyogre, rayquaza, ttar with smack down etc. All current and legacy level 3 raids are easy to solo (except for shuckle). Give us more challenging bosses, or boost their hp a little. A damage meter in raids. I want to be able to see how the damage I did compared to the damage done by other people.

- An easier way to switch moves between them. For example, TMs becoming available from PokeSpots, and CTMs no longer being random
- Some moves having additional effects. For example, Rock Tomb lowering Attack and dodging speed
- Pokemon to be able to learn more moves (not limited to 2 fast or 3 charge). Pokemon, when they gain new moves from updates, should add to the list of moves a Pokemon can learn instead of replacing moves (as a result, only community day movs should be legacy). This should only be done with an easier way to switch moves
- PvP system (as per everyone)
- Coin claiming (as per OP) - you should be able to claim coins without having your Pokemon knocked out of gyms.
- Being able to attack wild Pokemon with your Pokemon (as per OP)
- Being able to buy evolution items from the shop. I spent nearly a whole year waiting for 2 King Rock despite spinning a Pokestop almost every day. This may get worse with new evolution items being released in 4th Gen, and some of the new evolutions being highly desirable
- Being able to buy stardust from the shop
- Rebalancing some of the shop prices. It's ridiculous that buying 100 coins 5 times is cheaper than buying a bundle of 500 coins from the store
- Bottle Caps. Raid exclusive item. Can increase IVs
- Less glitches (as per everyone)
- Reduced animation for all that Lemywinks said, and also Pokemon evolution
- Getting stardust from gyms along with coins i.e. for every coin you get, you also get 100 stardust (it caps at 5000, much like how coins are capped at 50 - and you used to be able to get 5000 stardust before from gyms)
- Ability to buy tutor moves from the shop with coins - can be used to teach past community day moves for people who missed out
- Berry to increase stardust (as per OP)
- Ability to evolve Eevee non-randomly (costs 100 eevee candy instead of 25, but you choose the evolution)
- Can place friends into groups. For example you can make a group for overseas friends, and another for best friends

On CTM's --- allow the trainer to choose the move they want to teach so that it doesn't take 7 (!) CTM's to transition your Machamp from CC to DP...