For helpdesk ticketing systems we've used both OTRS and osTicket. We started out with OTRS and have since switched to osTicket. My lack of knowledge with perl and the complexity of upgrading was a deciding factor in switching. They both have their merits. OTRS to me seemed to have a lot more options available out of the box but again I was more comfortable with PHP and osTicket and made the switch there.
Inventory management if you have Windows (server) machines available take a look at Spiceworks or if you want something hosted elsewhere take a look at Meraki Systems Manager (they have a free version). I also liked iTop but ran out of time to take it slow getting it set up correctly/properly but for the 3 or so months we used it for a proof-of-concept it seemed to be great.
Check these guys out, they also have an online demo and a community version. And this book just came out 'Become ITIL Foundation Certified in 7 Days: Learning ITIL Made Simple with Real-life Examples'.. haven't read it though. In general, what you're looking for is ITIL stuff.. good luck.