Ohhh sorry I missed your point about price.
Shave cream and brush for $10-15 will be tough... but you can do it for about $20.
Couple cream options ($10 or less):
Palmolive For Men Classic Palm Extract Shave Cream 100ml https://www.amazon.com/dp/B0012Y1ELM/ref=cm_sw_r_cp_api_i_6sBwCb19FTMJ9
Old Spice Shave Cream - 70 g (Original) - Pack of 2 https://www.amazon.com/dp/000004458X/ref=cm_sw_r_cp_api_i_CvBwCbQZ9ART0
C.O. Bigelow Premium Shave Cream with Eucalyptus Oil 147g/5.2oz https://www.amazon.com/dp/B002O7UI18/ref=cm_sw_r_cp_api_i_DwBwCbW9Q5RHD
Proraso Shaving Cream, Refreshing and Toning, 5.2 oz https://www.amazon.com/dp/B00837ZOI0/ref=cm_sw_r_cp_api_i_hxBwCbMNJSWK1
And this is a perfectly good starter brush for $10:
The Body Shop Men's Wooden Shaving Brush https://www.amazon.com/dp/B001PDDEVS/ref=cm_sw_r_cp_api_i_CyBwCb3YSDEH8
u/amanforallsaisons did a fine job. However, you're asking about amazon and looking for a budget deal.
The Dorco PL-602 is the most-affordable good DE razor.
The Body Shop synthetic brush is a plisson-type synthetic. I don't have one, but I know people who do. It looks good, and should be soft-tipped, and easy to use.
EDIT: let's go with one I do have: razorock plissoft
Soaps are a littler harder - do you like fruity scents or non-fruity scents? There are shaving soaps on Amazon (Razorock and WSP) but not in sampler form. You won't go wrong with soaps from either Razorock or WSP.
A blade bank I picked this as the add-on item rather than Prime, because it's cheaper and you'll have enough to make the add-on order work if you use your cart instead of 1-click buying.
Get a sampler pack of blades. I prefer tryablade.com for samplers, but if you wish to do amazon, this isn't all bad. The normal problem with a sampler on amazon is that you're getting blades from the same factory, so you aren't really getting a chance to sample well. Here, they're at least well-regarded blades, so it's not terrible.
Alum - if you hold alum to your face where you nick yourself, it will stop the bleeding.
Witch Hazel. A lot of people like Thayer's. You can use any witch hazel you like as an aftershave - I put this one here because I'm tempted to get it myself. lemon Thayer's witch hazel
Balm. Not strictly necessary, but if you find yourself burning, it won't hurt to have on hand. nivea post shave balm
Honestly? If you use shave oil, like Shave Secret (walmart), you don't need the brush or shave soap. Savings can be had if you're incredibly price sensitive. But many people prefer the brush and lather method.
Shave bowl. I use a collapsible bowl which is convenient for bowl lathering when I travel.