Dear community!
I just bought an attiny13 (is the attiny85 better?) and wanted
to experiment with it.
Are there some hints how to programm these little fellows?
It would be very very cool if it would be possible to use the
Arduino IDE and language to do so. Or do I have to use some
other programmer and IDE? Are there some hints/beginner
tutorials etc. you can give me?
A little primer and I would be happy! 
Thanks in advance!
The ATtiny13 is one of the most limited of the tiny AVR family, 1k flash and 64 bytes (!) of EEPROM and RAM. It also has very limited I/O capabilities, lacking the USI capabilities of most of the other tinys (USI provides USART/SPI/I2C on tiny chips). Its main advantage is that it is very, very cheap (about 1 USD in single quantities.) Direct development is about your best bet with these little guys, as they are VERY resource limited.
Atmel's documentation and website are very helpful: http://www.atmel.com/dyn/products/devices.asp?category_id=163&family_id=607&subfamily_id=791 contains information about the whole tiny family. Take a look at the application notes on the ATtiny13 reference page here: http://www.atmel.com/dyn/products/product_docs.asp?category_id=163&family_id=607&subfamily_id=791&part_id=3175, that can be a great way to learn what these chips can do.
Regarding an IDE, WinAVR is a good choice if you're a Windows user and just getting started. I use emacs and makefiles on the Mac, it's pretty straightforward. There are a ton of tutorials available over at avrfreaks as well, google is your friend.
I'm quite fond of the ATtiny85 for small projects, it's quite capable for its size.
Added to the list...
http://code.google.com/p/arduino-tiny/issues/detail?id=8
I'd love to see an attiny13 core, them suckers are cheap!