Sorc 1/Cleric 1 should be pretty solid.

You'd have 3 L1 spell slots per day, you'd get access to medium armor and Shields, which is a nice boost to your AC. You'd also pick up the first domain feature for your cleric, which can be quite good. A S1/C1 would be a solid combination, it's a bit MAD as you'd want to make sure you have wis/cha/dex, but that's not the end of the world.

Sorc 19/Cleric 1 has a bit more issues. You only have the few L1 cleric spells, and lots of Sorcerer spells, make sure you took cleric spells that scale (so you can cast them in your higher level spell slots).

You miss out on the Cleric's Channel Divinity feature, which can be quite good. More importantly though, you miss out on the Sorcerer's capstone which will limit your effectiveness at L20. The Sorc's capstone is Sorcerous restoration which restores 4 sorcery points every short or long rest. Considering the number of things you have to use those on by L20, it would be a shame to miss out on that refresh.

A more significant dip into cleric might be more beneficial. 4 levels gets you the ability score improvement you miss by not taking Sorc 19. The second level of cleric gives you channel divinity and your second divine Domain feature. So I think, personally, I'd look at a heavier investment in your second class if you are going to give up your capstone.

The ring of mystic fire only affects the upcoming spell once

The ring of mystic fire (Magic Item Compendium 125) (7,500 gp; 0 lbs.) does, indeed, grant "a bonus to the amount of damage you deal with the next fire spell you cast before the end of your turn," but the ring only grants that bonus once. Each ray of a scorching ray spell isn't the spell; instead, the whole spell creates multiple rays. The ring improves the spell's damage only once by a maximum of +4d6 no matter how many effects result from the ring-modified spell.

(Ruling that each ray of the scorching ray spell benefits from the ring multiplies the power of the ring by the number of rays—with rather extreme consequences at low levels, especially given that the ring increases also the wearer's caster level for fire spells!)

Although the DM has several options available to adjudicate the ring's effect—allowing the player to distribute the total number of extra dice damage the ring grants among the rays before (or even after) the rays're launched, for instance—, this DM would keep things simple and have the player add the extra damage to his PC's first ray in a volley from the spell scorching ray, in much the same way that precision damage applies to the first attack in a similar volley (Rules Compendium 42). (In fact, I have a fire wizard in a current campaign who's eyeing this very item, and that's how I'm ruling it works.)

Hence, for example, under such a ruling, a level 6 caster wearing a ring of mystic fire can take a swift action to spend 3 of the ring's charges. Then the caster can cast scorching ray, creating 2 rays (because of the increase in caster level granted by the ring) and making 2 ranged touch attacks. If the first ray hits, it deals 8d6 points of fire damage. If the second ray hits, it deals 4d6 points of fire damage.