When the base year changes, does it affect data at current prices as well? I'm trying to collect current GDP data for India from 1990-2016. But, India changed its base year in 2011. So, to get a continuous series, do I have to change the data to a common base?
Yes, 

If you initially had GDP data at some base year $Year\;x$ for some $Year\;y$

$$I_{\frac{Year\;y}{Year\;x}} = \frac{GDP_{Year\;y}}{GDP_{Year\;x}} \times 100$$

If your base year has now changed to some base year $Year\;a$ for some $Year\;b$
, $a &gt; x$, then you should convert the GDP data from year $b$ onwards to reflect the initial base year, $x$.

$$I_{\frac{Year\;b}{Year\;x}} = I_{\frac{Year\;b}{Year\;a}} \times I_{\frac{Year\;a}{Year\;x}} $$

Further proof, 
$$\frac{GDP_{Year\;b}}{GDP_{Year\;x}} = \frac{GDP_{Year\;b}}{GDP_{Year\;a}} \times \frac{GDP_{Year\;a}}{GDP_{Year\;x}} $$

This way you will be able to get a continuous and consistent series based on some base year, $x$. If there were multiple base years, the data would be meaningless since it would reflect on changes with regard to different periods of time - ie, you would not be able to draw a definite conclusion.

An index compares the value of the current year against the value in the base year.
In this case, you might be looking for the current GDP value for every year from your data-set. As an example, if $$I_{\frac{2001}{2000}} = 110, \; and \; GDP_{2000} = 200, \; then \; GDP_{2001} =220.$$

When comparing data with changing base years, for eg,
$$I_{\frac{2001}{2000}} = 110, \;I_{\frac{2002}{2001}} = 105,$$
You can see that this does not mean the GDP in 2002 is $10\%+5\%$ higher than 2000. Instead, the actual GDP = 

$$I_{\frac{2002}{2000}} \times GDP_{2000} = I_{\frac{2002}{2001}}\times I_{\frac{2001}{2000}} \times GDP_{2000} = 115.5 \times GDP_{2000} = 231$$ 
or you can break this down to
$$GDP_{2000} = 200$$ $$GDP_{2001} = GDP_{2000} \times I_{\frac{2001}{2000}} = 220$$ $$GDP_{2002} = GDP_{2001} \times I_{\frac{2002}{2001}} = 231 $$ 
 Thus this is why you have to change your indexes to reflect on the same base year, since your calculated values of actual prices will differ.