After owning an Amiga 2500 for years, last year, I finally bought an A4000 Toaster. First thing you want to do is open the case and check the hardware. Toaster/Flyer systems are pretty specialized and you should check all the cables and connections--especially if you had it shipped to you. Then and most important is to check the battery. You might be able to look at it a bit without removing the Toaster card and if you can see it and if there is any sign of green or white fuzz on the battery, you will have to remove it immediatly and check the motherboard for damage. You will have to remove the Toaster card at least to do this and probably all the cards. If there is no battery damage, then you have a treasure to play with and the cool thing about A4000s is that you can easily upgrade them with a variety of inexpensive modern hardware. 4mb 72 pin simms are cheap if the MB memory isn't already maxed out. Vintage A4000s usually came with a 100-250mb ide drive which can easily be replaced by a 3-4gb drive which is also dirt cheap or if you are more ambitious, you can get an ide to compact flash adaptor and use a CF card as your hard drive. I use a 3.2gb ide laptop drive and a Norcent DVD burner in my A4000 and they work beautifully. I can't burn DVDs but I can certainly read data DVDs and also rewritable CDs which is handy for passing stuff from my PC or Mac to my Amiga.
There was a post recently about a 19 inch LCD monitor that syncs down to 15khz and sold on Ebay for $159. LCD monitors do wonders for Amigas and there are a few threads on Amiga.org about different monitors and ways of making them work with Amigas. :-)