I would recommend checking out the central market. It opens everyday from 8am until about 5pm and you can find locals shopping in large zepplin hangers and also little market stalls buying everything from smoked fish to pigs ears and beyond!
A little behind the market you will find the science academy, also know as stalin's wedding cake. You can take a trip back to the soviet era and enter the building and also visit the observation deck and see a marvellous view of the city. You can find it on Turgeòeva Iela.
A little beyond the science academy is the Black Market, if your bike is stolen in Riga you can generally go here and buy it back... ha! The market has everything and anything including soviet items and basically any wierd little thing you can think of that no one would want! You will find the market on Firsa Sadovòikova Iela.
This area is safe during the day but is best to avoid during the night times. If you are a keen photographer Little Moscow will provide the potential for some amazing shots.
The Moscow District is another district that is worth a stroll around, Apart from The Central Market (Covered in my shopping tips) with the Zeppelin Hangers there are also a few more interesting buildings like "The Academy of Sciences" that are worth a look.