Your best bet is to get another newer value machine and a decent graphics card. Install the graphics card in the value machine and you're set for most games. Or get the whole hog and go for a top of the line gaming rig.
1. Perhaps you should download and install the latest (last) video driver for your computer.
I had a similar experience with my vintage 2005 computer, after installing updated video drivers it was good to go! No lag on the low/medium settings of STO.
Perhaps if you have access to STO from another computer, set your STO video settings to minimum settings, then try logging on with your main computer in question.
2. For what it's worth, some "ancient" computers can run STO, such as my:
Lenovo Thinkpad T42 (2373-K5H):
~1 GB RAM
~64 MB video card (ATI Mobility Radeon 9600) with updated video card
~Pentium M 1.7 GHz single-core