I hesitate to offer up my perspective here as this topic comes up frequently and there are many established tiers that have strong opinions on the subject.
Still as a late starter and relatively new tier, say five years or so, I found buying a kit the perfect way to get started on what has since become a very enjoyable and expensive hobby. Not all of us had access to a mentor that would guide us along the way. I remember walking up to the tying materials wall at the local BPS and thinking "Holy Crap" how would you know where to start?
I purchased the upper end Orvis kit (about $200) when I started. Granted it's twice what the OP wants to spend, but this was my entry kit. I suspect the lower end kit is similar but limited. Again with full disclosure I have no affiliation with Orvis, merely this is where I chose to start.
In the kit I received all the basic tools required to get started. Have I purchased new and better tools since? Yes, but all the tools from the kit are still on my bench or in my travel kit. Moreover, the enclosed video CD, a Tim Flagler production, taught me how to use them.
The kit contained all the hooks and materials required to tie 16 different flies. It told me what hook to use, laid out the materials that were required, matched the bead heads to hook sizes, and introduced me to the terminology that related to the materials and how they were attached (hackle, dubbed, palmered, chenille, etc.). The sequence of the videos provided building blocks on techniques and complexity of the flies tied. Being an old military guy I would watch the video segment first, tie with the pause button at the ready, tie again, try without, and then review.
It taught me the basics of fly tying and left me with 160 flies, some horrible results, others quite productive. Some that have been fished out, others that I just never had any luck with (Chernobyl Ants come to mind).
I learned to tie thread bodies, palmer feathers, tie in zonker strips, tie parachute posts.
So for less than $1.25 a fly (that makes the basic tools free) I learned.
So from that starting point I now have three other vices, multiple bobbin holders, more materials than I can possibly use in my remaining days, and a understanding of what I need to purchase when I want to tie a certain type fly, and no regrets.
I strongly recommend that a new tier get a good kit as means of exploration.