Alright so I think I finished it for the most part with the exception of the last battle.
Here is a bit of a walk through for people that might be having trouble because of the Japanese.
Default ControlsZ is confirm, X is decline
A/D cycle some options on menus.
S is pause.
What to do?The game starts off with Yukari talking to you and then you get off a bus. Just chase the puppet and you'll get an event where you meet Marisa. A wild Reimu puppet appears and you take control of Marisa's Marisa puppet to fight it. I believe that this battle is a scripted loss. After you lose you will acquire the Reimu puppet and be brought to Sanae's shrine. You tell her that you come from the human world and want to go back home. She suggests that you to go to the human village. Proceed down and she'll come running back out again to teach you about catching wild puppets. After the conversation you are given 10 seals and can proceed. Just follow the path and you'll reach the human village. There is a cutscene right before you enter the village with Medicine but you don't have to fight her or anything. After entering the village go to the top center and enter the mansion. Talk to Akyuu in the middle room. I'm not entirely sure what she said but I believe it was about finding some sort of item to help you get back. Leave and you will see Wriggle outside. She flees to the right and joins Cirno, Daiyousai, Rumia, and Mystia. Walk over to them and a fight with Cirno will be triggered. After you win you will need to find where the rest of the bakas went (they're hiding in alcoves around the village edge). After you have defeated all five of them you will be able to access the area behind Sanae's shrine. Walk all of the way back and head up there. There are some fairies to fight and a ghost. You can find Reimu in the top left of the area. If you talk to her she'll acknowledge that you're training puppets but say you still have a long way to go. You need to then capture at least one of all thirteen different puppets. Once you have done so it will trigger a fight. Be aware that she has 6 strong puppets that are all at the trial's level cap.
Basic Puppet TrainingThere are thirteen puppets that can be obtained (at least that's all I could find).
You get Reimu as your starter. Route 1 has Kyouko, Seiga, Nitori, and Iku. The human village has Patchouli, Udonge, Alice, and Kasen. Behind the shirne has Rin, Kana, Aya, and Nazrin.
After a fight all puppets that are conscious will receive EXP and PP. Puppets of lower level will receive more EXP which will help to catch up. Whether they were engaged in battle does not matter. The PP are used to purchase stats boosts and new skills (attacks). You can choose to forget skills at any time but you will not receive a PP refund. After you hit Lv20 you will no longer receive EXP but you will still receive PP.

The options on the right there from top to bottom are view stats, swap order, give item, change abilities, and close.

First option allows you to put PP towards stat boosts. Second is to buy skills. You will be alerted when new skills are available after battles. Third and fourth options are to change ability and change style but neither are available in the trial.

Style: Normal (probably refers to Speed/Defense/Attack etc but changes are not available in demo)
Type: Sound
Nature: Green (green nature increases speed)
Bottom right is the hold item, this Kyouko has nothing.

The letter grades refer to the puppet's IVs. I have seen them range from E- to S. I'm not quite sure what the stats translate to. The first kanji means "come together" basically. The 24 is attack for that and the 36 is the defense for that. The next two have a kanji starting them that means the opposite "scatter". My Kyouko has 40 for that attack type and 27 for that defense type. I assume it's just kind of like the physical and special split in Pokemon but since there are no physical contact attacks in this game they came up with something else similar. The last one, the 39, is the speed stat. Just like Pokemon the one with the faster speed will move first. PP displays the PP the puppet currently has available to use for buying attacks/stat buffs. Last thing there is ability. Too many kanji there for me to tell exactly what it is but it appears to basically be Intimidate.
ItemsI don't have much info on the items since I only really used one of them the entire trial.

This one right here. This is the seal which you use to capture puppets. How capturing works is you first need to weaken a puppet so the seal will stick to it. After the seal sticks (you'll be able to tell if it sticks because they'll have a red ring around them) you need to finish them off. If you don't get the KO quickly there is a chance that the seal will fall off and then you'll need to toss out another. You cannot capture trainer puppets.
Unlike Pokemon you do not receive money from fights. Instead you will get items which you can sell. These items can be found under that red tab. The good news though is that wild puppets also drop items which can be sold for cash. You can find traders inside the healing centers and there is a shop with a ton of items in the human village.