To provide a 240V circuit, you must have a two-space, two-pole breaker.
A single-pole breaker obviously can supply only 120V loads, as it has only one terminal.
Single-space "tandem" breakers won't do because both halves will be on the same leg of your service, giving 0V difference between the two load terminals. Your dryer motor and light might work, but the heating element will do nothing. You'd also run the risk of overloading neutral.
There are some cases (mostly multi-wire branch circuits) where it is/was legal to handle-tie two adjacent single-pole breakers, but that's not relevant here. The dryer is a mixed 240V/120V load and requires common trip which is only available from a two-pole unit. It should be easier (and maybe cheaper) to just get the two-pole unit (instead of two singles plus a handle tie).
This is purely due to voltage, and has nothing to do with current. You can have two-pole 15A breakers, two-pole 200A breakers (or more), and many increments in between. It's true that single-pole breakers over 30A basically don't exist, but that's because delivering thousands of watts of power is more efficient at 240V (voltage drop is less of a factor), not because it's not possible.
Finally, "Home Depot" is not a breaker type. You must use breakers that are specifically listed for your breaker panel. Many will appear to fit, but will actually arc and destroy the bus.