There are a lot of things this could be. You might have worn shock/strut bushings, the ball joints, may be shot, and/or the tie rods may be shot. You need to look at this ASAP as anything other than worn shock bushings poses a serious safety issue to you and anyone on the road around you. You can use this as a time to replace other wear items like brake pads and shocks/struts at the same time. If there is any question, replace the part. You will save more than taking it to a shop and won’t have to redo part of the job in the future.
Yes, you can do this yourself. If this is your first time, block out a day to complete this job. You will need jack stands which you can pickup for $30-40 on sale. You will also need several tools for this job such as a ball joint press, tie rod/ball joint separators, a hammer, many different sizes of sockets, a torque wrench, snap ring pliers, etc. Many of these tools are available as “loan a tools” from parts stores. You buy the tool and they allow you to return it for a full refund when you are done.
If you feel you are in over your head, I would suggest taking it to a shop for two reasons. 1. You can get the car apart but not back together. You risk needing to have it towed or hire an onsite mechanic which would wipe out any cost savings. 2. If done wrong, this can pose serious safety concerns for you and others driving around your car.
If you decide to do it yourself, the first place you should drive after the work is the alignment shop. Depending on how well you do, the car may drive very poorly from alignment you just messed up and will definitely put excess wear on the tires. Get it aligned ASAP as there is no way to do this perfectly yourself.