How can I predict the upcoming weather for 1 week starting at June 3rd? Is there any way I can do this?

The instructions for this project give you a huge hint: air pressure. Since you already have 12 days of weather maps, you should be able to identify the following: 

Isobars depict air pressure (atmospheric pressure) on the maps. From this, you should be able to determine high and low pressure systems and cold and warm fronts. You should also be able to infer cloud cover (clear, partly cloudy, overcast) and wind speed and direction (barb points). Finally, you should be able to see air temperature, dew point, and barometric pressure. I'll explain how you can do this (and create your weather predictions) in two steps.

STEP 1:

Firstly, you need to make sure you have air pressure information, which will allow you to make a future weather prediction. Your professor referred you to worksheet 2, so take a look there. Isobars are lines on the map(s) that connect areas of equal pressure. The other information (cloud cover, windspeed/direction, air temperature, dew point, and barometric pressure) can either be inferred as your professor suggests or, if it exists on the maps, directly applied.

If you want to infer the above information using air pressure, keep this in mind:

High air pressure: less clouds, low precipitation, fair weather
Low air pressure: more clouds, high precipitation, rainy/stormy/snowy weather

If you have direct information - great! You can apply that in the next step.

STEP 2:

You will need to put your air pressure (and other) information together to tell a "story" about what might happen in a given area. Firstly, take a look at the isobars - wind direction is closely related to high and low pressure gradients. Wind will typically follow isobars from areas of high pressure to low pressure (that's one way to infer direction!), but can also flow from low to high pressure areas (that's why direction information is helpful). If you have direction information, use it to tell the story. If you don't make an argument for why the wind may move one direction or another. Hint: think about topography and bodies of water.

Wind direction can also tell you quite a bit about how clouds (and precipitation) are moving. If the wind is flowing from west to east, will it be bringing precipitation from a coast? On the other hand, will westerly winds be moving precipitation away from the area? Can you infer whether or not there may be rain/snow in you area? Next, look at wind temperatures. Is the wind moving from warmer to colder areas? From colder to warmer areas? If the area is getting colder and wetter, will it be cold enough to snow?

Your professor isn't looking for a spot-on 100% accurate prediction - that's tough even for the experts. What she/he is looking for is your line of reasoning. Why do you think the weather will be dry and cold, warm and moist, windy and rainy, etc...? She/he gives you a huge hint in suggesting that this information can all be inferred from isobars, so be sure to refer to your older assignments and/or textbook to make sure you have a firm grasp on that subject.