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.
The question and answers thus far focus solely on the role of the central bank. However, just like in our present fiat system, the government itself can also impact interest rates or intervene to affect monetary supply through the means of taxation (money destruction) and spending (money creation).
The following answer assumes a currency backed by a precious metal at a rate the government cannot change--no fair expanding the money supply by devaluing the coinage. This premise is ahistorical and unrealistic; governments changed the precious-metal content of their coins pretty regularly, sometimes even upwards. And even in spite of the government's best efforts at maintaining fixed precious-metal content, entreprising private citizens were always willing to step in.
But let's handwave all that away and see what a king can still do without banking.
If the government wishes to decrease the money supply, it merely needs to raise taxes, and then not spend the revenues. The amount of circulating money will thus decrease. (This is completely unsurprising to the MMT perspective on money--which argues that the same thing works with fiat currencies as well). Depending on period, many taxes may be in forced labor or of goods-in-kind, which somewhat blunts this strategy, but a strong enough state can decrease the money supply by just... taking the money.
If the government wishes to increase the money supply, there are two possible cases. If it already has a gold (silver, etc) surplus in excess of the circulating money, it can simply mint more coins and then spend them on things. As in a fiat system, it's the spending, rather than the minting, that matters; reshaping the gold in a vault doesn't impact anything other than carbon emissions, but increased government spending leads to increased demand in the private sector (from which the government is buying the goods it spends money on), and thus is a very effective stimulus to counteract recessions.
If it does not have a gold surplus, it could just borrow coins from the Italians and then spend them into the domestic economy. But more likely it'll need to find ways to collect more precious metals and do its own minting. Think the Thirty Years' War (the specific page referenced talks about the Spanish seizing private silver, but seizing gold in the form of artwork and plate were also very common) or Henry VIII's closure of the monasteries. The latter case is somewhat complicated because he has seized real assets of far greater value than the precious metals involved, but converting artwork into money is a time-honored way to increase the money supply.
Here the question sort of falls apart because of the reality that there wasn't a standard interest rate. The very nature of premodern markets constrains the scope of government action. Because the money supply has an artificial constraint (in the form of gold), and because communication networks were very much not instantaneous, the government could not function as a lender of any resort. Indeed, kings much more often found themselves in the position of desperate borrowers (I won't bother to link, but see the close interconnection between the Habsburgs and the Fuggers, or the nightmare scenarios in 1600s Spain or pre-Revolutionary France). You could argue that governments set rates by being a borrower of some resort, but again due to limitations with communication systems, you would be much more likely to see lending to the government by wealthy individuals or banking houses rather than the open-market bond operations of today, so government ability to set rates based purely on market participation would be rather limited. Moreover, given the sporadic nature of royal borrowings and the frequency of royal defaults on loans, it's a bit dicey to claim that loans to the government would constitute a floor or baseline "safe" loan against which all other pricing premia could be measured.
Instead, what you often saw was the government attempting to set rates--to zero, through bans on usury. This never really worked; people need credit, and they'd always invent some dodge for how to get around the legal bans, just as they did to get over the moralistic discouragement of lending.
Additionally, the existence of things like debt jubilees in medieval or ancient Mesopotamian practice surely had some impact on prevailing interest rates; if you know your loan will be wiped out on a 50-year clock or when a new king comes to power or whatever, that's going to affect things, probably to drive the rates upward. But I'm not expert enough on how these systems worked in practice to say exactly how. Also, this section of my answer has focused very much on premodern and early-modern Europe; I wish I could offer you more comparative data from a broader context.
Everything above has been looking at entirely domestic operations to manipulate the monetary supply within a particular country; however, the presence of Imperial Roman glassware in Han Dynasty China's tombs should effectively demonstrate that global trade networks are not new. In fact, one of the biggest use cases for actual physical precious-metal currency (not just as a unit of account) is foreign exchange. Far from mining being too expensive to contemplate, seizing precious metal resources was a major driver of colonialism and incalculably shaped today's world. Moreover, governments can readily intervene on behalf of private merchants to affect a more favorable balance of foreign trade and increase the domestic money supply thereby. These measures are obviously expensive (not to mention brutal and repressive for the targets of military aggression), but paid off well enough, often enough, that governments frequently made use of military conflict for monetary gain.
Even from a not-strictly-military perspective, governments were also ready to act as international market participants (such as the Portuguese royal spice monopoly, though there's a healthy helping of military conquest there too, or the Habsburg salt monopoly). Both these systems served to increase domestic money supply by a government directly engaging in profitable foreign trade (although the latter had a strong domestic component as well and could be regarded as a complicated form of taxation). This increased the domestic money supply by providing a non-tax-constrained way for the government to spend money into the economy.