I am interested in using a Cisco 1841 router to do some kind of load
balancing between two different WAN connections.  One connection will
be a T-1 and the other connection will be either another T-1 (from
another provider) or a business class cable internet connection.  I'm
not worried about true load balancing where every packet is load
balanced.  I'm also not worried about being able to use the aggregate
bandwidth during one session (i.e. download files at the combined
speed of the T1 and the cable connection).  I'm trying to increase the
overall bandwidth that is shared between about 20 workstations, so
that in general we don't have times when we max out our connection.

Can the Cisco 1841 handle this?  Are there other routers available
that can handle this?  Will configuring the router to do this require
technical expertise with the specific router being used?

Yes its do-able. Cisco 1841 should be able to handle this. 

When it comes to loadbalancing or loadsharing in this scenario, there
are 2 parts to it. Inbound packets and Outbound packet.

Outbound loadsharing is really easy. You can just have 2 default
routes on the router with equal cost and it will do outbound
loadsharing per flow(src ip dest ip src port dest port.

Inbound loadbalancing is tricky. If you use same vendor(ISP) for both
the links then its relatively easy, they can load share per flow the
inbound traffic. Downside you dont have ISP redundancy. If you have
connection from 2 different ISPs then only way you can achieve Inbound
load sharing is by getting your own IP address block and your AS #.
This would require little techinical expertise to configure.