Hello!
1) are you playing home or at a club/shop?
If you play at the local club/shop, you can use mat and scenery from there, you don't need to buy them.
If you play home, you need a mat and some sceneries. You can buy some kit (mat+scenery) from gw store, or you can make them by yourself.
I would suggest you the starter box (150€) only if you find it at its original price (150€, I don't know in $), otherwise it doesn't deserve.
Anyway, you need to buy 2 books:
- rulebook: core rules
- tome of champions: there are updates for cards profile, missions, campaigns, etc.. . This book will be edited each year in December, it is like the general handbook for AOS.
There is another book, which is "monster and mercenaries". It is not mandatory, but it is interesting if you want to add some mercenary (most are heroes) to your warbands, or you want to play in narrative way using monsters like chimera or giant spider from goblin.
2) do you like creating by yourself scenery?
If yes, you can build sceneries and a mat by yourself
There were 2 wave of warband cards release and some realeased in the white Dwarf magazine.
First: stormcast vanguard, Legion of nagash, goblin, ironjawz, bonesplitterz, daughters of khaine, idoneth deepkin, skaven, nighthaunt, flesh eater
Second: ogre, khorne, slaanesh, tzeentch, nurgle, slave to darkness, beast of chaos, sylvaneth, kharadron, ossiarch
White dwarf: fyreslayer, lumineth (#454)
Cards of the first wave are now sold out, but you can find them in the book "tome of champions"
As far as the second wave of warband is concerned, cards should be still available