(a) Let $n = 3$ and suppose that they intersect at a point $p$ and do not have the same tangent plane at that point. Show that $p$ is not an isolated point of $S_1 \cap S_2$.
(b) Let $n = 4$ and assume that $S_i$ are compact. Show that generically $S_1$ and $S_2$ will intersect in a finite number of points. This means, that for any $S_1$ and $S_2$ there exist perturbations of $S_1$ and $S_2$ such that the perturbed manifolds have finite intersection.
My attempt for this part: To the contrary suppose $p\in S_1\cap S_2$ is an isolated point of $S_1\cap S_2$. Then $\exists$, a neighborhood $V$ of $p$ in $S_1$(can also be in $S_2$, but a similar argument will work) such that $(V-\{p\})\cap S_1\cap S_2= \emptyset$. Then, locally around $p$ we have $S_1\cap S_2=\{p\}$ and $T_pS_1=T_p(S_1\cap S_2)=T_pS_2$ which is a contradiction to the fact that the intersection don't have the same tangent plane at $p$.
I know implicitly somewhere I am using the fact that $'n=3'$ but I don't know where.
For the second part, (b) Suppose the intersection $S_1\cap S_2$ has inifinitely many points then as $S_i's$ are compact, any infinite set has a limit point say 'q' and so does $S_1\cap S_2$. Then, any neighborhood of q(either in $S_1$ or in $S_2$) has infinitely many points of $S_1\cap S_2$. Moreover, the tangent spaces at $q$ doesn't span $\mathbb{R}^4$, but then I don't know how to get a contradiction to compactness?
For part (a), if you have two different tangent spaces at $p$ coming from $S_1$ and $S_2$, then you know that the intersection at $p$ is transverse. I'll let you look for it, but you should have some theorems by now that tell you that (in this case) you can parametrize a one dimensional neighborhood of $p$ in $S_1\cap S_2$.
For part (b), you would use other theorems to conclude that any intersections can be perturbed to be generic - i.e. made transverse. Once this is done, what can a transverse intersection between two surfaces in a 4-manifold look like? Think about this a bit, but you should conclude that they must intersect in points. (In fact, whenever you have two manifolds who "fill up" the dimension of the ambient space, they will intersect in points. In this case, 2+2=4)
Lastly, why must we only have finitely many points? There are some assumptions we haven't used yet. Let me know if you need any more hints.