I understand that under a fiat currency, the government can buy securities on the open market to increase the money supply, and decrease interest rates. This monetary policy is possible to help recover from recessions. could this be done with a gold standard? Could the government increase the supply / lower interest rates?
Yes, in principle, under a gold standard, the central bank can buy more gold, or build a mine and mine it, which increases money supply. This lowers the short run interest rates. But the issue is that it typically can't fight the market price of gold: when the price of gold increases beyond the parity, people rush to exchange their paper currency for gold at the bank. This happens until money becomes so scarce that ir regains its value relative to gold.
This is very different from what happens with fiat currency. With fiat currency, the central bank can buy bonds, or gold, but it can even just print money and give it away. ( The act of giving it for free to individuals is called a "helicopter drop". Dropping it from a helicopter is one way to actually do it...)
Even though in a gold standard, the central bank can increase money supply, this is different than buying bonds. This is because since money is backed by gold, it can't loose value relative to gold. In this case, it's really hard for the central bank to lower the value of money if it wants, for example, to stop a deflation. Instead, because bonds are also denominated in the country's currency, making money lose value also makes bonds lose value and vice versa.
In a theoretical gold standard where there is a 1:1 correspondence between money and gold, and private sector debt is strictly controlled (so that it does not become a money substitute), the central bank is highly constrained. It would have very limited ability to do much of anything in terms of policy.
In the real world, various "gold standard" systems generally only featured partial cover of money by gold. (There were various systems over the years; the post-war Bretton Woods system was distinct from the inter-war gold exchange standard.) In this case, so long as the central bank stays away from minimum gold cover requirements, it had freedom of action to operate in the money markets. Assuming other countries remain on the gold standard, the only thing that is fixed is foreign exchange rates (that is, the central bank cannot intervene to raise/lower the value of its currency).
If a country was facing a run on its gold supplies, it lost its freedom of action. It either had to raise rates to attract gold inflows, run austerity policies to reduce demand and hence imports, or break their peg to gold. In the 1930s, countries got knocked off gold one by one, as the policies to maintain the pegs were politically unsustainable.