From what I gather:
1) If I am doing a one-sided test then the table I choose (normal distribution or $T$ table) depends on the sample size. If sample size $\geq 30$ then I use normal distribution table; and if the sample is $< 30$ I use t table. Is this correct?
2) If I am doing a $2$ sided test then I always use $T$ test table with degrees of freedom being $\infty$ for large very large $N$. Is this correct too?
What determines if you should use the $t$-distribution or the Normal in your hypothesis test is whether or not you know the variance of the random variable. If you do not know the variance and instead use an estimate of it, you should use the $t$-distribution regardless of if it is a one-sided or two-sided test.
Having said that, as the sample size increases, though, the $t$-distribution converges to a standard Normal which might justify using the Normal if the sample size is large enough.
More precisely, by the Central Limit Theorem, $$\frac{\bar{X}−\mu}{\hat{\sigma}/\sqrt{n}} \to^d \mathcal N(0,1)$$ as $n\to\infty$, where for a random variable $X$ with mean $\mu$, $\bar{X}$ denotes a sample mean, $\hat{\sigma}$ denotes the sample estimate of standard deviation of $X$ and $n$ denotes the sample size.