Most important: 40mm ruler. 15mm scale is played with 40mm wide bases, which are conveniently the unit of measure (UD in the rules). These are not converted and are an important part of how the game is built. Whatever the unit frontage, both sides need to be the same and the unit of measure needs to be that as well. Again, for 15mm the frontage is 40mm and this is a long and well-established standard that many gamers have been using for decades. I recommend Litko as an easy source for rulers. They have a variety of DBM/DBA/FoG/ADG gauges at 40mm. This (http://www.litko.net/products.php?product=40mm-Linear-Gauge#.WYSPZ4qQxdA
) is popular among the players and works well for many purposes. I have it and it is grand. I even more highly recommend this set (http://www.litko.net/products.php?product=DBA-3.0-Compatible%2C-Small-Scale-Gauge-Set-%285%29#.WYSPyIqQxdA
). It comes with a 5 UD, 4 UD, 3 UD, & two 1 UD devices. That covers a lot of common measures and lets you put down a perimter with one while measuring movement with another (for example, staying out of the 4 UD shooting range of archers while moving your unit up, or aligning the deployment zone distances from side and base edges).
Next most important is materials to represent terrain. ADG is a rank-and-file game at a somewhat abstracted level. Most 15mm FoW terrain will be completely inappropriate for use in it (for example, you do not fight in individual houses. A village is an area of terrain on the board which applies a given set of modifiers to those inside/fighting in it. The markers you put down to represent the terrain type are like individual trees in FoW, they are moved out of the way to let the action happen). Additionally, terrain pieces have minimal and maximal sizes as well as some restriction on shape. For instance, fields are supposed to be broadly rectangular and not too blobby, while forests or marshes are intended to be a bit less regular and more blobby. I recommend getting a few pieces of felt from a craft store and cutting them out into legal sized shapes (the book lists the sizings in the terrain section; I am pretty sure the minimum size at 200pts is 2x3 UD, but do not take my word). This is very cheap and easy and is a great way to start out. After playing a bit it becomes easier to understand what sort of fancier terrain will work with the system and one can consider buying/making more involved items. I use mostly felt and decorate it with stands of trees, buildings, etc. to make it look a little better and represent the typing. One last thing, cut a few different sizes of each type. Terrain choice and placement is a very important part of the game and makes a big difference for many armies, particularly at higher level play. You may want small pieces in one game and big ones in another so it is best to have both (this is particularly true for your Samurai with their combo of good horse and medium foot).
The third thing you need is a d6. I trust you have at least one of these lying around somewhere. Actually, I find having 8 of them is ideal with at least two distinct styles. When setting up I lay out 6 in the sectors of the field and then when I roll to place my terrain I use 2 different styles of dies to make the sector and positioning roll simultaneously. The extras are completely optional. Only a single d6 is actually required to play. More just speeds things up a little here and there.
Finally (I think), the last thing is damage markers. All units take between 2 and 4 hits from shooting and fighting before destruction and these usually accrue in a stepwise fashion. What you chose to use is up to you so long as it is easy for both you and your opponent to to recognize which units are damaged and how many hits they have on them. These tokens usually take on the form of something layed down behind the damaged unit. Although, some folks use napoleonic casualty caps. These are unpopular because they can bend you spears and bows, etc. if not placed/removed carefully. Gary uses little cardboard cutouts that read 1, 2, 3, or D(isordered horses by elephants or camels). I used colored glass beads for a while with green, yellow, red signifying 1, 2, or 3 hits. But have recently taken some of my spare figures and put them lying down on washers to represent hits. Mike uses FoW tokens when he does not use my indicators. I have seen people use a wide variety of other ideas. A couple use poker chips. One had little colored flags. Another had laser cut MDF with hit numbers. My advice is to use an internally consistent method that is easy for your opponent to follow. Some folks had trouble following my bead method, so I decided to switch (I actually really liked the idea of actual casualty markers already and just used that as a push to do a batch up). I also recommend that you keep your makers in the 20-30 mm width/diameter range. smaller can be hard to see and the token might not get moved with its unit. More and it becomes hard fit them together if you have a line of damaged units and occasionally can get confusing as to which unit it is assigned to. I made a couple of downed horsemen for two-hit markers on some larger washers I had to hand. The washers are at least 35mm in diameter, which I think ruins some of the effect and utility when placed behind a unit.
I think that about covers it. A game mat is a good idea (particularly because the field is not a standard foot size), but is not really essential to start out. A lot of the local types already have a mat so while learning you should be pretty well off using your opponent's. When you are ready to get one, a sufficiently large cut of felt from Joanne's cost me about $35, which included enough extra to cut some small terrain shapes as well (although being the same color as the mat makes them of marginal use without painting or putting atop another piece). I would assume that you could assemble all the accoutrements you need, even by buying a set from Litko, for ~$40, maybe less. Even if you sprung for a game mat immediately and spent on some nice 6mm terrain decorations (I strongly recommend 6mm. 15mm terrain items look really weird at the approximate ground scale. Units of 6-8 men represent 100-300 men so standing next to a FoW church is...awkward. Plus 6mm is inexpensive and easy to transport) I doubt you would burn more than $100 if you tried.