Should I replace the roof or just fix it

My other answer only answered "what should I put under the shingles". I reread the question and see that @Vebjorn asked how to install shingles. Doing a shed is a good project if you haven't shingled before. Doing a house is harder because of the peaks and valleys.

To shingle your shed, assuming a straight gable roof:

A. Roofing felt.

1. Buy a roll of roofing felt: you can get that at Home Depot or wherever.
2. Starting at the bottom of the roof, staple down a layer of roofing felt. Just roll it out, then use a staple gun to fasten it at the top, bottom and middle about every 2 feet along.
3. After the first layer is put down, add a second layer. Overlap the first layer by about 6 inches.

B. Shingles

1. Start with a layer of upside down shingles, with the tabs cut off. Ask the person at Home Depot for a knife for cutting the shingles - it'll be so dull and gross afterwards you won't want it for anything else. Position the shingles so the overhang the sides and leading edge of the roof by about 1/2". Nail the 1/2 shingles with the strip of tar at the base of the roof, putting nails in the shingle 4 across, top and bottom, i.e. eight nails per shingle.
2. Measure the width of the uncut shingle. Measure that length from bottom of the 1/2 row of shingles on each side of the roof and make a mark. Snap a chalkline between those marks.
3. Do the first "real" course of shingles. Again, overhang by about 1/2". Put 4 nails per shingle, about 1" above the cuts that define the tabs. Put one nail on each side, and one above each tab cut. Use the chalkline to keep them straight.
4. Now the second course. You want each shingle to be offset by 1/2 tab from the shingles below, and up by 1/2 a shingle width. On the sides of the shingle are little cuts: bend the top part and that will give you a reference to "butt" against the lower shingle. That will keep your shingles level. Likewise, there is a cut on the top of the shingle, in the middle of the tab. Bend that up on the first row of shingle (already nailed to the roof). That will give you a place to butt the new shingle side-to-side, to ensure that the new shingle is well placed horizontally. After the first one is placed you can just butt the shingles against each other, so only the tabs on the new shingle are necessary.
5. When you get to the end, cut off the flap as best you can straight with the line you want. Then go back and do the gap at the beginning: you can often use the cutoff from the end you finished to fill in the end you started. (It probably wasn't clear in (4), but I like to start with a full shingle, then come back and fill in the bit at the beginning of the course).
6. Continue until you get to the top. The last shingle should fold over to the other side. Then do the other side.

C. The peak

1. Now you have two sides of shingles folded over. You need to finish the peak.
2. Cut a shingle into three tabs, cutting up-and-down at the tab slot.
3. Turn a tab sideways, and fold it over the peak at one edge of the building. Choose the edge away from the prevailing wind. It should be half-and-half on each side.
4. Measure how far down the peak it goes, the mark the same distance from the peak on the other side. Snap a chalk line.
5. Continue to nail tabs (1/3 shingles) along the peak, two nails per tab, using the chalkline to line them up. You won't have the tabs to help you with the offset now, but by now your eye will be pretty good. :)
6. Finish up with a 1/2 tab. Tar the exposed nails.

Replace it now. Like TODAY. That's BAD.

With that kind of bad shingle wear, your shingles are likely to create ice dams which will cause water to back up and leak through the roof, causing interior damage, compounding the cost.

In the summer, the roof won't hold the water because it can't freeze, so the curls will only grab a very little water before it drips free.

In the winter, however, a little retained water will freeze, expand, and capture more water, which will freeze, and so forth.