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The absorb elements spell says

The spell captures some of the incoming energy, lessening its effect
on you and storing it for your next melee attack. You have resistance
to the triggering damage type until the start of your next turn. Also,
the first time you hit with a melee attack on your next turn, the
target takes an extra 1d6 damage of the triggering type, and the spell
ends.

At Higher Levels. When you cast this spell using a spell slot of 2nd
level or higher, the extra damage increases by 1d6 for each slot level
above 1st.

So if my Sorcerer is hit with a fire bolt from the enemy mage, I use my reaction to cast absorb elements at 3rd level so I now have 3d6 fire damage stored for my next melee attack, I then use my action to cast a 1st-level spell and my turn ends, resetting my reaction. Another enemy hits me with chain lightning and I use absorb elements again at 3rd level.

Do I now have 3d6 fire + 3d6 lightning or just 3d6 lightning stored for my next melee attack?

No, the extra damage is only added to one attack on your next turn.

Read the Absorb Elements text carefully:

Also, the first time you hit with a melee attack on your next turn, the target takes an extra 1d6 damage of the triggering type, and the spell ends.

(emphasis added by me)

The spell only provides a benefit during the turn after you cast Absorb Elements. You cannot "store" the extra damage across multiple turns.

If the Sorcerer did not make a melee attack roll on their next turn, then the benefit of Absorb Elements is lost at the end of that turn.

Note: It is possible to cast Absorb Elements twice such that the durations overlap, but the damage won't stack. This is because a character's Reaction resets at the start of their turn, not at the end:

When you take a Reaction, you can’t take another one until the start of your next turn.

For example, suppose the Sorcerer takes fire damage before their turn, using their reaction to cast Absorb Elements. Then on the Sorcerer's turn, once their reaction resets, they somehow take fire damage again, and spend their reaction (again) to cast Absorb Elements (again). This would mean having multiple castings of the same spell, but their benefits do not stack:

The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect--such as the highest bonus--from those castings applies while their durations overlap, or the most recent effect applies if the castings are equally potent and their durations overlap.

If the Sorcerer proceeds to make an attack on this same turn, then only the "most recent" or "most potent" casting of Absorb Elements would apply to that attack.