This is very much a need. The Turtlebot is a low-cost ($1300) way to get a lot of functionality out of the box, though the lack of manipulator does limit what you can accomplish.
If you (or your lab/company) does have a five-figure budget for hardware, there are a number of very attractive off-the-shelf options which exist to prototype on: http://wiki.ros.org/Robots/#Portals
(Disclosure: I work for Clearpath Robotics, which design and manufacture several of the machines on that page.)
That's what membership in your local fablab is for.
Give it 5 years though and I can see home kit manipulator, teachable robot kits that you could program to maybe water the plants or dry some dishes.
There is the hexbug spider which makes a nice, hackable beginner platform (http://www.amazon.com/HEXBUG-Hexbug-Spider-colors-vary/dp/B0...) with the addition of an arduino nano.
The kilobots are also cool (http://www.eecs.harvard.edu/ssr/projects/progSA/kilobot.html), and an even smaller, cheaper robot (http://www.hizook.com/blog/2011/09/08/infrared-remote-contro...) using the same principle of locomotion.
You can buy inexpensive robot platforms, but they're small - there's a reason for this. Heavy, powerful components cost real money and someone has to make margin selling them to you, given they paid margin to buy them. High power and high current control systems (motor drivers, motors, etc.) are more difficult and expensive to design than small, low-power units.
But, there's nothing stopping you from seeing a design you like, and making it from the raw parts, except a lack of tools (in most cases).
> a simple little robot platform moving around my house sensing tape on the floor. Even that will cost me $1000
Huh? You can buy line-following robot kits on Amazon for $30, and there are even cheaper ones out there.
> When can I build a robot that will perform real, useful automation that will improve my life, even in some marginal way, not just seem like a cool toy? How can I build a robot that will fetch me a beer or let the dog out or get the mail or something even minor that will make me feel like I am doing something with purpose and a real goal and not a homework assignment to get a robot to perform some meaningless task.
Right now, buy some motors, buy some metal stock, bearings, shaft, pulleys, etc. and start designing a machine around them. Long before you get to the algorithms of running a robot, there are the physics of it, the basic electronics knowledge, and the design and machining.
> What are some robotics projects that I can build and learn from while and actually use and enjoy, today, relatively cheaply?
Small robotic arms are cheap these days - OWI and Lynxmotion make fairly inexpensive arms for learning, small wheeled bases like the Dagu Wild Thumper can carry a small arm for around $250, and are well regarded. You can throw an Arduino or Raspberry Pi in for control put together a Lynxmotion arm and a Dagu base, and for less than $600, you have a platform that can lift small objects and move them around. When you want bigger, make it yourself from what you've learned.