Without knowing the Slutsky equation and income/substitution effect, how can I show a certain good is inferior or Giffen?
yes. Notice that a good is normal, inferior Giffen for a certain value of prices/income. For example, a good might be inferior at some prices/income but normal for some other values of the prices/income.
In some cases, however, you can make stronger statments. For example, homothetic preferences always lead to normal demands as
$$
x_1(p_1, p_2,m) = x_1(p_1, p_2,1) m 
$$
where $x_1(p_1, p_2,1)$ is the unit income demand function which is always greater or equal to zero.
From this, it follows that homothetic preference can never lead to Giffen goods (but to show this, you do need the Slutsky decomposition).