How do I fill an object with a solid color with lines through it?

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Yes, as Larry suggests.

1. Select your  shape, in the appearance panel create a new fill above your solid color fill.

2. In the swatches window, click the upper-right corner icon to expand swatch list.

3. Choose "Open Swatch Library / Pattern Swatches / Geometric Patterns / Lines

4. Then choose the swatch that most looks like the effect you prefer.

NOTE: You may need to rotate the pattern 45º to get it to look like the example. In this case, select your shape then select the top fill with (the one with the lines) in the appearance panel. Next, double-click the rotate tool and set it up to rotate 45º and uncheck all tick-boxes except "Pattern".

ADDITIONAL NOTE: It may be the case that the Illustrator patterns all consist of black lines and not white. There are a fiew different ways to get the pattern lines to be white. I would begin by seeing if a blend mode might fix the issue. If this doesn't work, I would then drag the pattern swatch from the swatch panel and onto the artboard where there is no artwork. This should allow you to edit the swatch. Change the black lines to white and then drag the new swatch — clipping box and all — back into the swatch panel.

Hope this helps.

You would need to add a new fill to the object in the Appearance panel and use a pattern fill from one of the pattern libraries. I don't have AI on this machine to show you the steps. Sorry