I have done some basic wiring on 120v circuits, adding outlets etc. I want to add two 30 amp 240v circuits to this panel (w/dual pole breakers). Its a 200 amp box with 100 amp breaker going to a subpanel and a 40 amp breaker for the oven. My question is about connecting the neutral wires to the same bus bar as ground. It looks to already be set up this way, but I'm just looking for some feedback on this plan.
I wouldn't recommend working in a live electrical panel unless you are very experienced and understand the considerable hazards. Especially in this situation where there is no obvious upstream overcurrent device. Slip with your screwdriver and you could have a pretty serious fire, arc explosion or electrocution.
I highly recommend consulting with a electrician licensed in your area. Not just for the task of adding the circuits but also to investigate why there is no upstream overcurrent device/disconnect. (edit: ThreePhaseEel reminded me of the "Rule of 6", where a panel does not need a main breaker if all the power can be turned off in 6 switch flips or less. This installation may be correct under that exception.)
If you are worried about cost you may be able to reach an agreement with the electrician where you do the majority of the work, with them just doing the final hookups. You'll stay safe, and guarantee a safe installation.
As to your question you cannot know if it is correct to connect the ground wire to the same terminal bus as neutral without knowing more about the overall electrical system. Neutral and ground should be connected (bonded) in only one place, usually the main panel. If this is the main panel, ie the two heavy gauge wires in the lower middle come straight from the utility or meter, the installation is probably correct and you can wire additional circuits like the range circuit is wired already. If this is a subpanel (perhaps in an apartment or multi-tenant situation) the panel may need to be corrected to separate neutral and ground.
My biggest concern is I don't see the wires to the ground rods or water pipes.
It looks like this is the "service" panel. The biggest clue is almost not identifiable in the photo, on the right side of the panel between the 100A conductors there is a dark spot. I think this is a screw head that attaches a short piece of metal that is a jumper that bonds the metal enclosure to the neutral bar.
At the service for multiple reasons the neutral is grounded, mostly to limit the voltage to ground on the hot legs. (People may argue other reasons are more important, they could be right, it's not worth arguing about.) This neutral bond renders the ground and neutral essentially indistinguishable in the service panel to the point that the code doesn't require separating the wire termination in a service panel. After the service the all neutrals and ground are normally required to be maintained separately.
What does bother me is an essential component of limiting the voltage to ground is an actual connection to "grounding electrodes", which generally means connecting a #6 or larger copper wire from the service ground bar to ground rods and water pipes. I just don't see those wires in that panel. I wouldn't work in that panel unless I knew the voltage to ground from either hot conductor was limited to 120v. It should be grounded at the utility transformer, but without an actual grounding electrode at your service you're relying on a connection you can't see.
Back to your question, if this is the service it appears it was installed as a 6-disconnect main panel. Adding two 30A 240v circuits would probably be OK, I can't see the actual size on the wire feeding the panel, but guessing it's a 200A feed your total load couldn't exceed 200A, you should be fine. If the feed is less than 200A an inspector could require a "load calc" that didn't exceed the feeder size. Even if the simple addition of breakers exceeds the feeder size the load calc has diversity of load considerations, and you likely won't be prohibited from adding the additional circuits.
If you’re unsure on how to perform this work I strongly suggest you hire a licensed electrician.
If you insist on doing this yourself contact your hydro authority, apply for a permit, and have them pull your meter base. Then do the work yourself and have it inspected by the hydro authority or an approved delegate such as a master electrician.
You get only one life. Don’t risk doing this work in an unsafe fashion.