The first thing you need to do is to learn the cards and the Yaku. This is all explained by McM on the Hanafuda Rules Page here: http://www.animecubed.com/billy/bvs/partyhouse-hanafudarules.html
Now that you've done that, what you need to learn is the priority of each card to get. The goal being to grab the highest priority cards you can each round and get Yaku as quickly as possible.
The biggest scoring Yaku is the Brights, which gets up to 10 points for all five of them. 3 of the Dry brights are needed to score 5 points, the other Dry adds 3, and Rain Man adds 2. This means that the Dry Brights are a high priority to grab, but Rain Man isn't such a high priority since he can only increase the score but can't be used to get the 3 Dry Bright Yaku. In fact, Rain Man will probably be at the bottom of your priority list unless you already have multiple Bright cards.
There are two 2-card Yakus for 5 points each, The Loop and The Sacrifice which are unique to Billy-Style rules. The Loop needs Full Moon and Lightning, and The Sacrifice needs Full Moon and Lily by the Bridge. This puts Lily by the Bridge and Lightning just under the Brights in priority (higher if either player has Full Moon), and catapults Full Moon to the top priority spot.
The next priority slot goes to the three 3-card 5-point Yakus. Poetry Ribbons, Blue Ribbons, and Boar/Deer/Butterfly (aka RoShamBo in Billy-Style). You want to get these if possible and prevent your opponent from getting them if in any way feasible. If you have one and your opponent has one, then the third drops to being just another Animal or Ribbon card.
Next on the list is the Juice Cup, which counts as both an Animal and a Plain card. That means it is two cards for the price of one! Sweet!
After Juice Cup comes every other Animal and Ribbon card. You need 5 Animals/Ribbons (each is a separate set) to score one point, with 1 extra point for every card in the set over 5 you have. There are 9 Animals and 10 Ribbons.
Last are the Plain cards. A whopping 10 are needed for a single point, but there are 25 in the deck (counting the Juice Cup) and every one past 10 gets you an extra point.
Your card priority probably looks like this now:
Full Moon
Other 3 Dry Brights
Lightning/Lily by the Pond
Blue Ribbons/Poetry Ribbons/Boar-Deer-Butterfly
Juice Cup
Any other Ribbon or Animal card
Any Plain card
Rain Man
Now, there are factors that can raise and lower a card's priority, so let's cover those now.
Raised Priority:
Card is on the field
Card is the last card needed for a scoring set for yourself
Card is the last card needed for a scoring set for opponent
Card will increase your existing score (If your next play will get you more points, you might really consider calling Koi Koi, more on that further down.)
Lowered Priority:
Card will not aid you in scoring (Basically, if there is another card that will bring you closer to a Yaku. If you have four Animals and no Brights, go after the Animal card and leave the Bright alone.)
Card is 'guaranteed' (This is what you call cards that your opponent has no way of claiming. If you have two cards of a month in your hand and the other two are on the field, those are 'guaranteed' cards. You aren't in any hurry to claim them. This also refers to if you have one in your hand, another on the field, and the remaining two have already been claimed.)
Card is 'dead' (This is rare and only happens when three cards of a single month are dealt to the middle of the table at the beginning. This creates cards that neither player can claim.)
Strategies while playing the game (barring cheats):
What do you do if a high priority card is on the field and you have no matching card in your hand?
Suck it up and go for the best play available to you. If you can't match a card, you can't match it. Pray that you get a match on the draw or that your opponent can't match it either.
What do you do if you opponent grabbed a low priority card on the field that matched a high priority card in your hand?
If there is no other match on the field, hold on to the card as long as you can. If the match hits the field, rejoice in the feeling of a guaranteed card.
I have nothing in my hand that matches anything on the field! What now?
It's time to do some damage control. Check your hand. Check what cards your opponent has claimed. Either play a card from your hand that matches another card in your hand and pray your opponent can't grab it, or play the card that helps your opponent the least if they do grab it.
I have a scoring hand! Should I Koi Koi or Bank?
Ah, the big question. To Koi Koi or not to Koi Koi.
Here's some questions to ask yourself when considering Koi Koi:
How likely are you to score more points if you continue?
Scoring more points is the only reason to call Koi Koi at all, so if it doesn't look like you'll score more points within one or two plays, then go ahead and Bank the hand. If you have something big that you can get on the very next play, then really consider Koi Koi, but check a couple of other things first.
How likely is your opponent to score points if you continue?
If your opponent has called Koi Koi, then don't even ponder this question, just Bank the hand. If your opponent is within: 1 card of a 5-point Yaku, 2 Animals, 2 Ribbons, or 4 Plains from a 1-point Yaku, then calling Koi Koi is probably going to be a very bad idea and you should Bank the hand. Don't worry if it's a low score. Remember, a low score is better than no score.
How likely is the hand to end before any points are scored?
If your opponent has diddly and you have one card left to play and a low-scoring hand, you might consider going ahead and playing the last card for a chance at another point or two. However, if you have a high-scoring hand it is probably better not to risk it.
Playing the block:
The defensive game becomes more important at harder difficulties, especially when you have a bad hand.
Sometimes the best thing you can do is slow your opponent down.
A high priority card, such as the Full Moon is neither in your hand nor on the field, but another card from the same month is on the field and you have a match in your hand.
Playing the block in this case means grabbing the card so that your opponent has nothing to match the high-priority card to if they either have it in-hand or end up drawing it.
Your first priority in a defensive game is to keep the position of dealer, even if it means scoring only a single point on the hand. However, the second priority is to keep your opponent's ability to score points low.
Grab cards from 5-point sets to deny those options to your opponent. Try to make sure that if they do win, they can't do so with a score of seven or higher.
Cheating:
If you're confident about winning without resorting to any cheats, then don't cheat. Getting caught loses you cards, which is painful, and winning without cheating gives a nice 20% bonus.
However, cheats can be indispensable. Particularly at higher levels of difficulty. All cheats are best done early in a hand since you can't use them after someone Koi Kois, and the damage to yourself early on is minimal (you can't lose cards if you haven't taken any after all).
If you plan on cheating, then stock up on Sleight of Hand levels. If you're going to cheat anyway, you might as well get more chances to do so.
Momentary Reflection and Future Viewing are pretty bad and not worth the loss of the 20% bonus or possibility of loosing cards. by the time when you really should cheat (hard/very hard) there are much better cheats around. also at hard and beyond your opponent can redraw his hand rendering future viewing nearly useless at that point.
False Accusation isnt bad however you get this at level 5 which is when Med Hard unlocks. if you are playing on Med hard know that they have a 50% chance of catching you. its better to wait until Level 7 comes around when you can spin the wheel.
Spin the Wheel is fantastic. Getting rid of a bad draw-hand and getting dealt a brand new one can turn the tide completely.
False Confidence can destroy your opponent if you play it right. Making them Koi Koi the next time they get a scoring hand gives you an opportunity to pull the whole hand out from under them. Another trick with this is using it before you Koi Koi yourself. This can give you an extra chance to get a card or two to make the 7pts you might need to double your score and limits the risk of them stealing the hand from you.