Rebel Sport NZ sale. Don't feel qualified to comment on other non-tennis sports, so here are my picks.
Cheap Balls
Grand Slam 3B $5
It's cheap tennis ball. Never played with it and don't intend to. Probably worse in quality than the expired Head Championship balls, but your dog won't know the difference, probably.
Tour Comp 4B $6
Actually cheaper/ball. Still an ultra budget/beginner/dog ball.
Cheap Grips
Tecnifibre Overgrip 3P $7.50
Tecnifibre makes nice grips, I haven't personally tested this, but I have the replacement grip version of this and it works nicely. $2.50/overgrip is a great price.
Tecnifibre Feel Grip $11.24
These are really good replacement grips, nice and plush. I've seen these in Australia go for ~$16 AUD.
Tournament grade balls
Tournament grade are so much nicer than "championship" or "club" class balls, because are more highly QC'ed and have a more consistent bounce, ball to ball. All balls in this category apply as tournament grade. Tournament balls are usually ~$20 NZD for a 4B can, which is just plain daylight robbery. These are superior in quality to the Head Championships advertised in the other deal.
US Open Ball 4B $10
US Open is an okay ball. For the price it's good because it's still a tournament class ball, but I'd go with the other two if you can spare an extra $2.50.
Tecnifibre XOne 4B $12.50
Same ball is $20 at Thump Sports. I've reviewed this ball extensively. Good bounce, doesn't fluff up. Lettering doesn't last very long but the internal ball pressure is very good even if the outside looks worn out. Ask for full review.
Dunlop AO 4B $12.50
The official ball of the Australian Open. Reasonable tennis ball that's stocked by my tennis club. I personally haven't tested it properly.