Can you cast a spell using mana from a land that was just played this turn?

When you play a land, it enters the battlefield and you may use it right away to produce mana — as long as its mana ability simply requires tapping it (or paying other costs). Summoning sickness (CR 302.6) prevents a creature from attacking or using activated abilities with the tap symbol ({T}) or untap symbol ({Q}) in the cost, but basic lands and most dual lands are not creatures, so summoning sickness does not apply to them at all.

Mana abilities are special: they don't use the stack and resolve immediately (CR 605.3a). So tapping a freshly played land for mana is perfectly legal. The mana produced can then be used to cast a spell or activate an ability in the normal way.

The only exception would be a land that is also a creature (e.g., a land animated by a pump spell, or something like Mutavault while activated). If such a land-creature entered the battlefield this turn under your control, summoning sickness would prevent you from tapping it for mana via an activated ability using {T} (CR 302.6).

Concrete example: On your first turn you play a Forest. You immediately tap it for {G} and cast Llanowar Elves (which costs {G}). This is completely legal — the Forest entered the battlefield this turn, but that doesn't stop you from tapping it for mana.