Do I need to upgrade my service/panel to be able to have a ground?
OK first, the panel has a lot of "double-stuff" breakers (2 breakers in one space or 4 breakers in 2 spaces).  

The breaker in 3/4 (right side, top 2 spaces/4 breakers)

Take a careful note of the double-stuff(s) in spaces 3 and 4.  Treat those 2 breakers like one quad breaker, keep them together or better, tape them together.  Move them together as a unit.  Note the red wire.  That red wire has a partner black, and that partner must be above the red wire on that same "quad breaker".   This will put them on opposite poles, which is essential to not overloading the neutral.

Further, that quad/double-double cannot be there.  Look at the drawing on the panel sticker; it's a fair bit confusing, but your panel has 12 spaces in 6 rows, and each row has a black dot on the sticker drawing.  The bottom 4 dots have the thing where it splits into 2 breakers (e.g. 19 and 20) but breakers 3 and 4 do not have that.   That is telling you double-stuff breakers are not allowed in the top 2 of 6 rows - exactly where breaker 3/4 is.  

So you need to take the bottom 2 right-side breakers out (the normal ones), and depending on available wire length either move the quadplex/double-double that we've been discussing to those 2 spaces, or move the whole column of breakers down.  Then, refit the pulled normal breakers in positions 3/4.   That will take care of the "double-stuffs in illegal locations" problem. 

NEC 1965 legislated "CTL" rules that breakers and panels must have keying/locking to make that mistake impossible. But it typically takes 2-3 years for NEC revisions to get approved by state, so it probably slipped in under the wire. Or they used non-CTL breakers which were prevalant at the time.  GE used CTL to totally redesign how they do double-stuff breakers; so you will not be able to source a replacement for these double-stuffs. DO NOT use a BR, HOM or Siemens here!!! 

Time to think new panel.

But you don't need to replace the whole panel.  You can fork out to a subpanel, keeping the original main breaker and using this existing panel as a hub feeding the subpanels.  That lets you DIY the subpanel work without having to deal with the fact that the wire before the main breaker is always-hot.  

Not least, this panel is laughably small for any home; I like to see homes have 40 spaces.  If you are adding a subpanel, I like to see around 48 total spaces, so a 40-space or two 18s would be a good addition.  If you have any aspirations to a generator or a modern on-power-failure power system, Siemens is a particularly good choice due to cheap generator interlocks.  There's no point staying in GE since few of your breakers will fit a modern GE panel. 

Another strategy is to put a bigger subpanel positioned (with consultation with your power company) so they could easily put a larger service drop on the new location. In case you ever want to upgrade electric service. 

There is no limit on subpanel size; you can use a 225A subpanel with 100A main service.  The subpanel must be >= the breaker feeding it. 

So that is what I would do with this panel; slowly convert it to simply a feeder to subpanel(s).  You'll need something to fill the empty holes as  you move branch circuits out; just use the old breakers for that. 

All subpanels connect 4 wires: two hots, neutral and separate ground from the main to the subpanel. Grounds and neutrals are separated in the subpanel. 

Grounding

First, your panel needs a ground bar of its own. YOu might try contacting an electrical supply house that is a GE dealer and see if they have a ground bar that will bolt up to this panel in existing holes.   All grounds should go to the ground bar, including the main Grounding Electrode, and then there should be a heavy wire (heaviest that will fit in the holes, copper) connecting the neutral bar to the ground bar.  It's a nice bonus if you can get a clamp meter around that wire; it's very useful for diagnosis.  

Now, your panel will need a grounding electrode system.  That should be a typically bare wire from the grounding bar to a water pipe, ground rods, or an UFER ground buried in the concrete.  Water pipes are fallen out of favor because of the water company's penchant for replacing them with smart meters made out of plastic.  Now two ground rods are required (unless you pay for a test that costs more than a second ground rod). The UFER ground is still the best option, but it must be planned before the foundation pour.