A "needle" is properly called a "stylus". The cartridge is a small "block" that the stylus is inserted into. The cartridge is then usually screwed onto the tonearm. The cartridge will have some small pins at its back end which will attach to wires running through the tonearm. You attach or detach the wires to the cartridge with a small pair of pliers. The wire ends should have little metal sleeves that slide over the cartridge pins.
Many cartridges have a replaceable stylus. If your cartridge is one with a replaceable stylus, all you need to do is buy a replacement. The odds of your stylus being replaceable, or if it is replaceable, the odds of finding a replacement stylus for your particular cartridge, are pretty well nil. You will probably have to buy a whole new cartridge/stylus assembly. This would be prudent given its age anyways.
In buying a new cartridge, I would agree with what Bignerd100 said. Get a cheap Grado. Good value for the money and an appropriate choice for your table. Have your local audio shop set it up, ...if there is one near you, ...and if they know how to do it. Otherwise, you can buy one online at a place like AudioAdvisor and install it yourself.
I am not familiar with your particular table, so my comments will be generic on turntable set-up.
Essentially there are four things you need to do.
First, level the turntable. If the table has a suspended subplatter (i.e. the the table is bouncy within its frame), there are probably some screws to adjust level. Adjust to level and then put the whole table on a level surface. If the table does not have a subplatter, or if it cannot be adjusted, just place the whole table on a level surface.
Second, if the tonearm is height adjustable, adjust it so that the tonearm is level with the record surface when it is playing a record. You would adjust the height by loosening a screw where the tonearm is attached to the turntable base, then lifting it up or down, then retightening the screw. If there is no height adjustment possible, well, there's nothing to do. You take what you get.
Third, the really tricky part. When you install the cartridge to the tonearm, usually with screws, the cartridge needs to be at the right angle. You cannot do this accurately by sight. You need to have a cartridge alignment gauge. It's a flat piece of paper or plastic or metal that you set on the turntable platter. It has markings on it against which you line up the front of the cartridge. Some gauges are expensive, but some can be purchased for only a few dollars. If you can't find one, or if you don't want to pay what the available ones cost, then align your cartridge so that the front surface of the cartridge is parallel with the front surface of the tonearm. In other words, use the tonearm as the gauge. This will get you by, but you really can't do a proper job without an alignment gauge.
Fourth, adjust the stylus pressure. Typically, this is done by turning the counterweight at the end of the tonearm. The weight settings are usually marked on the counterweight itself. Or there may be a small dial on the turntable base underneath the tonearm. You can buy relatively inexpensive gauges to measure the weight more precisely, a Shure SF-2 stylus force gauge for example. However, you should be able to use the turntable's gauge without having to buy a separate gauge. It would be recommended to buy a gauge if you upgrade your table if you set it up yourself rather than have a shop do it. The instructions with the cartridge will tell you what the stylus force should be for that particular cartridge.
Turntable set-up is a disappearing art. And like Bignerd 100 said, you only need to practice it a thousand times for it to become easy.