What to do if time series are non-stationary? Data: 
I have a time series data of 2528 daily observations for OMXS.30 (Stokholm) closing price.
The aim is to fit proper ARCH/GARCH models and use for forecast daily Value at Risk. Here is a plot of my data, and the daily log returns. 

By looking at the log returns, can I assume the mean is stationary? (I've seen others do that but I'm not sure it's a valid assumption.)
My question is, what do I do if my data is non-stationary? Is it supposed to be?
If I've understood it correctly the time series is non-stationary if the ACF decreases just as it does in the (series data) plot below. Also , the number of significant lags in the ACF of logreturns are many. What does that mean and how do I continue from this point?

As you've probably understood I am VERY new to all topics of this project, and I've never dealt with time series before. Any help would be very appreciated!
 A: Take a look at my comments What "more" does differencing (d>0) do in ARIMA than detrend? suggesting alternative approaches ( best suggested by the data ) to  evidented non-stationarity symptoms. Classicly differencing is more approptraite for stock market data.
Often times power transforms like logs are attempted to deal with non-constant error  variance When (and why) should you take the log of a distribution (of numbers)? . A viable alternative in many cases is Weighted least Squares as suggested by Tsay to deal with stock price data . See page 13 of https://pdfs.semanticscholar.org/09c4/ba8dd3cc88289caf18d71e8985bdd11ad21c.pdf to follow this thought.
A: We actually do not need statistics itself to look at stationarity, and the first test for that is common sense. There is no stationarity because there are external factors that have a major influence on the data. For example, the market crash of Sept. 29, 2008 (in gray in the figure below), and the Dow Jones Industrial Average over that period of time.

From https://www.macrotrends.net/1319/dow-jones-100-year-historical-chart'>Dow Jones - 100 Year Historical Chart
Certainly, these two markets (Dow Jones and OMSX) are highly correlated. So it is likely more of interest as to which outperforms which than trying to predict either one from auto-correlation. Also, note that Dow Jones has a futures market, so that its futures market may predict the OMSX index futures better than it can be predicted from its own data.
I would, as a first step, detrend for external influence, e.g., look at OMSX/Dow Jones, before examining autocorrelation or other trending. For example, one potential way to make money might be take advantage of any reliable ARIMA time delay between the markets. Finally, analyzing this type of data has a lot to do with what the objective of the analysis is.
