I have an early 80s Trek 970 Single track that is an awesome bike, but as an older rider who doesn't ride trails as much, would like to ride in a more upright position. The handlebars are for the most part straight across and am just looking for a way to increase the height of the handlebars to make the ride a little more comfortable.
You need either a shorter (stem)[https://bicycles.stackexchange.com/a/270/19705] (brings the bars closer) or to raise the stem.
The age given strongly suggests its a quill stem, and will look like this on the inside.
Back off the clamp bolt (may be a hex socket or a bolt head) a couple of turns, and then gently tap down on the top of the bolt to loosen the clamp down inside your head tube. Don't undo it fully unless its seized with rust and you have to get grumpy.
Once loose, pull the stem up a bit and retighten. Go for a test ride and see how it feels.
WARNING Be mindful of the minimum insertion marks and obey them.
Once you're happy, tighten up the clamp bolt properly after checking the stem and front wheel are in line. Use a ruler - don't just eyeball it. Also avoid overtightening the clamp bolt.
If adding height doesn't help, you'll need a shorter stem, or possibly bars with a little more swept-back and some more uplift. This starts costing money though.
Depending on your willingness to invest in new hardware, there are a couple things that can be done if there is not enough range in the quill length to make you comfortable.
You can get a new stem that has more 'rise'. For instance, a zero rise stem looks like the diagram above, getting a 5 degree, 10 degree, etc... rise will further increase the bar height.
You can invest in some riser bars... the riser bars are measured by the rise from the clamp. A straight bar would be no-rise, but there are several pretty affordable rise options from .5" - 4" based on your preferences.
If you consider either of these solutions be sure to know your bar clamp diameter, based on age of the bike it is likely 25.4mm but understand that sizes there are a number of different bar clamp sizes for mtbs and road bikes.