Finding some rational points on elliptic curves 
If I am considering an elliptic curve, for example 
  $$y^2=x^3-2$$
  $$\text{Edit: and } y^2=x^3+2$$
  over $\mathbb Q$, how to find rational points?

What possibilities do we have to calculate some of the rational points on it? Are there even possibilities for calculating integer points on the curve?

The simplest way is to use existing methods in computer algebra systems, e.g. if you use the online Magma calculator here there are now awfully sophisticated algorithms there for this sort of thing. To learn more you could read the relevant section in the Magma handbook here
In the case of the first of your curves, if I put in the following

E:=EllipticCurve([0,0,0,0,-2]);
MordellWeilGroup(E);
RationalPoints(E : Bound:=1000);

then the output is

Abelian Group isomorphic to Z
Defined on 1 generator (free)
Mapping from: Abelian Group isomorphic to Z
Defined on 1 generator (free) to Set of points of E with coordinates in Rational Field given by a rule [no inverse]
true true
{@ (0 : 1 : 0), (3 : 5 : 1), (3 : -5 : 1), (129/100 : 383/1000 : 1), (129/100 :
-383/1000 : 1) @}

