I'm just not sure what the pattern is for using each, or if there even is one.
I suppose that you are talking about cases like:
I want to go home, and he wishes to stay at the club.
The computers are growing much faster every year, and new models have more memory then the old ones.
You need to use а in cases when you wish to switch your attention from one event or fact to another (1) or to stress that two events contrast in some way. That is, one of the circumstances (3) or the whole situation of the second phrase (2) contradicts to the first event. For example:
- Я слушаю новости по телевизору, а он читает их в Интернете.
I listen to the news on the TV, but he reads online newspapers.
- Я знаю ответ, а ты -- нет.
I know the answer and you don't.
- Мама уехала на дачу, а папа -- на рыбалку.
My mom went to dacha, while my dad went fishing.
On the other hand, you would use и to stress the temporal (ex. 1 below) or causal (ex. 2 below) relations between the two events, or to say that the two events both support the same idea (ex. 3 below):
- Я слушаю новости по телевизору и потом проверяю их в Интернете.
I listen to the news on the TV and then compare them with the online newspapers.
- Я списал ответы у Паши и сдал экзамен.
I copied the answers from Pasha and passed the exam.
- Мама уехала на дачу, и папы тоже нет дома.
My mom went to dacha, and my dad is not at home either.
However, it is possible to have contrasting situations joined by и, as in (ex. 1 below). These cases are different from the contrasting situations joined by а in that they usually sequences of events. One uses а when one talks about general facts (ex. 2 below), habits (ex. 1 above) or simultaneous events (ex. 2 and 3 above):
- Он видел смерч и остался жив.
He saw a tornado and lived to tell the tale of it.
- Он любит загорать, а не купаться.
He like to sunbathe, not to swim.
I don't think that my answer is complete, but I think that it captures the general idea.
I actually wrote a big blog post about this a while ago.
To simplify the matters for "и", it is a coordinative conjunction. You'd use it to put together two parts of a sentence which are of equal grammatical strength. It just links one with the other. The English equivalent would be “and”.
Éве стáло скýчно и онá откры́ла Youtube. (Eva got bored and she opened a Youtube tab.)
Contexts for "a" are numerous, but can be narrowed down to three:
a) revealing or stressing the difference
Все засмея́лись, а я не пóнял, что бы́ло смешнóго в том, что я сказáл. (Everybody laughed, but I didn’t get what was so funny in what I had just said.)
b) revealing or stressing the likeness
На дворé стоя́л декáбрь, а за окнóм крýпными хлóпьями пáдал снéг. (It was December, and it was snowing heavily.) Here both parts of the sentence add up to a harmonious picture. Here, “a” is easily interchangeable with “и”.
c) stating the facts, often stressing neither difference nor likeness.
У неё мать фи́нка, а оте́ц га́нец. (Her mother is Finnish, her father is from Ghana.)
2. Connection, the “conjoint a”
a) describing a picture, or a course of events
Напро́тив на́шего до́ма раски́нулся тени́стый пáрк, а за пáрком разлило́сь ди́вное о́зеро. (Right across from our building there is a wooded area, behind it there lies a picturesque lake.)
b) describing a turn of events
Послы́шался стук в дверь. Открыва́ю, а там сосе́д по ле́стничной площа́дке. (Knocking at the door. I opt to answer — there stands my neighbor from across the hall.)
3. Deviation from the norm, “a of unusual consequence”
Ле́то в Ки́руне необы́чное: на часа́х два но́чи, а на у́лице светло́ как днём. (Summers in Kiruna are quite unusual: the clock says 2 am, but the sun is shining as if it were noon.) The emotional connotation here is surprise (at the weather).
There is a major difference between these two words:
и is соединительный союз ("connecting" conjunction) and means and, for example:
вчера и сегодня
yesterday and today
а is противительный союз ("opposing" conjunction) and in meaning is closer to but in a sense that it requires two opposing elements, e.g
я - тут, а ты - там
I am here, but your are there
Note that in this example, the English sentence could use conjunction while:
I am here while you are there
In addition to this, a can have another usage: in sentences like
- А давай пойдем в кино!
- А не пойти ли тебе на...?
- А это что такое?
In these cases a is not used to oppose two elements but is standing on its own, usually in the beginning of a sentence, more as an exclamation, expression of emotions, which can be translated as what if in most cases, but can sometimes be translated as and in a similar usage:
- What if we go to a cinema?
- What if you just f... off?
- And what is this?
In most situations, the distinction between "и" and "а" is clear: "и" means "and" (adjunction), while "а" means "but" (contrast, comparison, or discrepancy). There is, however, a situation where "а" means "и". Consider:
Сначала у нас зарядка, а после зарядки завтрак. (First, we do the morning exercises, and then we have our breakfast.)
Там стадион, а за ним универмаг. (There's a stadium, and a department store behind it.)
In both cases, using "и" in place of "а" is OK:
Сначала у нас зарядка, и после зарядки завтрак. (First, we do the morning exercises, and then we have our breakfast.)
Там стадион, и за ним универмаг. (There's a stadium, and a department store behind it.)
However, there are cases when you must use "и" in very similar situations:
Мы сделали зарядку и позавтракали. (We did our morning exercises and had our breakfast.)
Там стадион и универмаг. (There is a stadium and a department store there.)
Я обернулся и увидел её. (I turned around and saw her.)
Since there is no "symmetry" here, I would recommend using "и" for "and" in all situations, at least in the beginning.