What is the fastest way to increase my company's value?
A combination of the above approaches seems to be the best route to success.
I found a useful display in the Company menu, which breaks down your current company value by category. Essentially everything that you buy and produce, plus all cash that you currently hold, and minus any bonds, contributes to your value - but the highest contributors for me (so far) are:
Businesses - these are all places that offered me to buy them out; I then connected them up to my network and reaped the profits. I didn't bother paying towards any auctions of unprofitable businesses, but if I'd had the ready cash, this would likely have been viable too. The value of businesses fluctuates depending on your rail network.
Shares - while buying out a competitor is indeed an expensive prospect, the value of my shares contributes to my company's value. So buying up shares while they're cheap is worthwhile, as those shares will increase in value along with your competitors. Whether this is better than just hanging on to the cash, I'm not sure. The value of shares fluctuates depending on your competitor's rail networks.
Buildings, Bridges, Tracks and Locomotives (which are all related to building a rail network) are obviously the main contributors. What's interesting is that more expensive rail lines, such as those involving tunnels and bridges, contribute more than cheaper options. It goes without saying, though, that if you've got a large network, it will have a high value. The value of these items fluctuates depending on your rail network.
Cash contributes as much as everything else - and it's handy to have a sizeable amount spare, so you can invest in any opportunities that arise - but in the long run it's better to invest it in network expansion, shares, pretty much anything that can net a profit or increase in value.
Here I am, not far off the target - just gotta pay back that bond and I'm good to go! :)