Usually, the best option to increase any skill from a very low level is to go to a trainer, as you have a very low chance of using the skill, which means less skill increases. If you're in the mages guild, and you've done Ajira's quests, she should be at a 100 disposition, train with her. At you're alchemy level, it should be dirt cheap.
That would be my recommendation to, while you CAN make potions at low levels, since you'll fail a lot until you have a decent alchemy skill, it's easier to train the skill instead.
Though, just because you can't see the effect of a potion doesn't mean you can't use it. If you drink a potion, it effects you regardless of whether you have enough alchemy skill to use it or not, it's just that you won't be able to tell what effect the potion has, though you can usually judge from the name, like a cheap restore health potion has a restore health effect that's weaker than a standard restore health potion, also, drinking potions has no chance to fail, though I'm pretty sure it can't request your alchemy skill. With just eating ingredients directly, on the other hand, though it can increase your skill, it's unlikely to have any effect on you at low skill levels, a strange design choice, I must, since in real life you don't need to know the nutritional facts of a specific food item to absorb whatever is in it, but hey, that's the way the game is designed. You can also still make potions if you can't see the effects, as you can sill combine ingredients with the same effects, you'll be able to tell that you'll be able to make a potion if at least one question mark appaears in the effects list and there is a potion name. The main problem is that since you can't see the effects of your potions, you can only try random combinations of ingredients until you find ones that will work, unless you remember some of the effects a specific ingredient has. You'll also fail a lot, and since in Morrowind, skills only improve from using them successfully, that means your skill may level slowly. You're potions also won't be very good, but that's not much of a problem if you just want to increase the skill to raise intelligence, it's only a problem if you actually want to use your potions or make a profit selling them.
Anyway, as far as I understand, none of the available factions have any actual quest conflicts with the temple, except maybe the Imperial Cult, logically, they should conflict, but since I've never really tried to join both with the same character as I figure it makes little sense to do so anyway, I'm not actually sure. But in any case, there are some factions that the temple might not be fond of, but I don't think joining any of them will keep you from joining the Temple, it might just make people in the Temple not like you that much.
Now, there is a certain point in the main quest which will cause people in the Temple to not let you join, advance or recieve new quests, but after a certain point, I think they accept you again, I won't say more on that here.