Welcome! I just started a 32 biocube although mostly using old rocks from my main tank.
Algae blooms are very common for new tanks and can take months to resolve. Even though I’m using well established rocks I got diatoms on sand and glass. Generally takes between a few weeks to a few months to get all that stuff sorted out. There are a few more annoying types of algae that may cause issues.
What does the algae you have look like?
As for lights, really depends what you intend to keep in the tank. If soft corals and lps, then likely the stock light is sufficient. If you want to go the sps route (and you should likely wait a few months before adding sps), then you should upgrade. I was debating between a kessil and ai prime and decided to go with ai prime mostly because I got a good deal from a local store and wanted to somewhat keep on the lower budget with this tank.
You may consider a skimmer like Tunze Comline, note that for almost any skimmer you’ll have to make some adjustments to the back to fit it. I’ve decided to go with water changes for nutrient export for now but likely gonna get a skimmer soon. Also, In Tank makes much better media baskets for the biocube which also allow to have enough room for a skimmer.
Most people seem to run chemipure blue, phosguard and filter floss. I had some extra from an old QT tank so put it in, but otherwise plan to just run BRS Rox carbon. The best advice I can give in this hobby is to be very patient and make sure to allow ample time for almost anything and never make sudden changes unless it’s an emergency to save your livestock.