Clan Takeda starts with a split empire. They have 3 territories in the west on the underside of the Mori Empire (Aki, Bingo, Bitchu). The other three territories lie in the highly contested eastern side of Japan (Kai, Izu, Sagami).
Koku is a problem from the beginning for the Takeda Clan. The average harvest will yield a little over 1500 koku a year. This isn't that bad until you realize you need a sizable army to defend your split empire. However, Aki has silver deposits and Kai has gold deposits so that has the potential to add and extra 1000 koku per year, which is great.
One thing you are given to help you with the split empire setup is 2 ports, one in Aki and one in Sagami. The ports add 200 koku a year each to your income. They also allow you to transport troops over seas between your territories in the west and the east. This takes the pressure off of needing to build troop production facilities in the west right off the bat.
To help the army, Takeda gets to recruit and maintain cavalry cheaper than the other clans. Cavalry is one of the most powerful units on the battlefield. The Yari Cavalry are the fastest unit in the game which is good for running down routed enemies. Cavalry archers make great scouts and Heavy Cavalry are the equivalent of tanks.
The tough part about playing Takeda is the starting position. You have a split empire and no defensible territories. Every one of your territories has a hostile enemy clan on its border. Your starting army contains Cavalry which no other clan starts with so you have some power advantage there but you hardly have enough koku to build any more. Even with your reduced cost, Cavalry are not cheap. Your yearly income is not that spectacular to support the size army you will need.
As always, start by trying to sure up some of the negatives just discussed. First is the split empire problem. You can try to maintain your entire empire if you want but I would not recommend it. So you should consolidate one of your 2 empires into a smaller area and just maintain it while you expand on the other front. Which to expand and which to consolidate is a tricky question in itself. However, capitalism should prevail in your thinking and you should choose to expand in the east and consolidate the west. The Mori lands in the west are worth 100+ koku each. This is not spectacular and will not even cover the cost of one of your Yari Cavalry units. So take your troops from Bitchu and move them to Bingo. Once Bingo is attacked move the troops there to Aki. This will be the western outpost you keep alive. It has a silver mine and Iron and Sand deposits for an armory.
Now you must take care of the koku problem facing your clan. Lord Takeda is a skilled general and his men are very powerful. Needless to say the best defense is a good offense when it comes to playing this clan. So attack and take what you need to survive. The best territory to take is Musashi without a doubt. It has the highest koku output in the game for a single territory. It is also a river province. This means 2 things: 1) it is very easy to keep once you have it and 2) it will be very hard to conquer if you don't do it in the first season. So there it is, take Musashi in the first season no matter what it takes. Take in all your force from Sagami and an extra Yari Samurai from Kai. Hojo defense in the start of the game is very weak. So do it early and your set.
Now that you've had a battle you can see how good Takeda is on the battlefield. You've already taken Hojo's most valued territory and it was well worth it. So take more of Hojo now. By taking Hojo out early you can avoid the Hojo Horde. If you don't know what this is then play Shimazu or Mori and then you will. You are now on the border of Hojo's capital province of Shimosa. Take it next and soon. By taking out Shimosa you cripple Hojo's troop producing facilities. This means his small army will be left high an dry with no reinforcements. Even if you can't hold Shimosa take it anyway. Because Hojo will have to spend seasons rebuilding his Dojo's in order to produce reinforcements. You will be producing enough troops to take him out while he's doing this. Thanks to adding Musashi to your territory.
What to Build:
Now that you have all but killed Hojo, you need to start worrying about infrastructure and better troops. What I find to work very well is to build Naginata in Aki since it's your only province capable of it at the moment. You can also produce your Yari troops there too and they will have an Armor bonus. This is good since you are probably going to start feeling pressure from Uesugi who loves his Samurai Archers. In Kai your capital, build your Archers and Cavalry for now as well as your special units for spying. If you get the Legendary Swordsman event by now and you should of, then you want to set up shop for No-dachi Samurai. Put it in Shimosa. The reason for this is that it is on the coast and there for you can install a port and be able to ship them far away. Just a note, don't bother with Christianity. When the Portuguese arrive just turn them away. Wait for the dutch so you can get Muskets right away if you choose to use them. I always include one or two in my army just for emergencies. You are now on your way to building a very nicely balanced army, which can rip any one apart.
What About Shinano?
Shinano seems like the most tempting target especially for Takeda since it gives an honor bonus for Cavalry and it has Iron and Sand for an armory. Don't go there yet. Shinano has 10 surrounding provinces. You will be attacked constantly for control of Shinano if you take it early. Take Shinano once you have at least 4 or 5 of the surrounding territories. This way the number of enemies on your border is reduced and easier to deal with. Once you do take it, set up your cavalry shop there. Horse and Spear Dojos as well as an Armory, an Archery Dojo too if you like to use Cavalry archers.
Now it's up to you really depending on the board in front of you. If Uesugi is not to over powering finish him off and take his lands. At that point you will pretty much have all of the most koku rich lands in Japan and nothing should stop you. Don't forget about your wonderful western front from the beginning. You can fortify your eastern front and then start shipping troops over to Aki in order to start crushing Mori or whomever is in control over there. This tactic will allow you to surround whomever has control of central Japan and attack them from the east and the west. Or you can simply brute force it from the eastern front and take every territory one by one all the way across the board. It is not that hard to do with the powerful Takeda army and generals.
The situation around you at this point may be different than what it was for me so it's hard to say what to do next. Keep these points in mind.