Okay, as "johnznothere" said, the Pokeblock Case (which you want and/or need - get it ASAP) is in the Contest Hall in Slateport City. If you're not sure which building it is, it's the one with a Blue roof and Red walls in the top-left corner of Slateport. All 'Contest' buildings look like that. (There is a Contest Building in Verdanturf, Fallabor, Slateport and Lilycove cities.)
The "Mixer" they're referring to is inside the building, and you'll notice that there are 2 grey table-shaped blocks. This is the Berry Blender. One of them will have nobody beside it, and the other will have an Old Man standing beside it (and maybe even another 1-2 people as well).
The Blender which has nobody using it is so that you can connect your GBA/GBAsp with (up to) 3 other People's GBA's, and do BerryBlending (or 'spin the bottle', as I like to call it) with your friends.
However, not everyone has friends that play Pokemon RuSa, so the old man (and the other 1 or 2 people) are there so that you can do it without having to link up.
What you do is put in a berry (Higher numbered berries tend to make better blocks, but I've not tried it with #36 and higher, and _don't even bother_ with #15 and below), and as the little red thing comes around to your spot, just press the A button.
If someone misses, then it slows down, if someone gets almost right, then it speeds up a bit, and if someone gets it exactly right, it speeds up even faster.
After a while, you tend to get the hang of it, and with my friends we've made a Gold Pokeblock - Level=62, Feel=33.
Now you're thinking - what's Level and Feel? The Pokeblock's level represents how much a Contest Condition will improve - Higher=better
Feel represents how full it will make your pokemon. After a while they won't eat any more Pokeblocks, so Lower=better.
Personally, I use my Low Level/Feel pokeblocks in the Safari, and the Pokeblocks with High Level/Feel (Like the Gold Pokeblock I mentioned before) ratio I use for contests.
One more thing to note- if two (or more) people (in a link Blending) use the same berry, the mix tends to go bad, resulting in a Black Pokeblock (they have a VERY low level, and are pretty much useless - I discard them instantly). Also, I try to make sure I have at least 5-10 of any berry before I use it for blending, so that I can easily plant more without waiting forever to get a few more.
Hopefully that has helped you a bit.
