The Deckbuilder Toolbox is... honestly, it's a bunch of bulk. Which is fine if you want to play casually, but if you want to play non-Commander in Constructed formats like Standard or Modern, they're a waste of money. It's almost always better to buy singles for building decks-- and in most cases, singles for the most part usually aren't going to cost more than $1 a piece. In fact, for a lot of decks in Standard, the money is focused in maybe 10-15 of the 75 cards.
That said, you can build pretty nifty budget decks for either format, too. There's a lot of solid budget decks to be built in Modern that are really no more than top-tier decks with some of the extremely expensive cards replaced for cheaper options. In fact, my Affinity deck (aka super cheap robot synergy) probably doesn't cost more than $100 all told and I've come in 1st at Friday Night Magic with it. I'm personally a big fan of
SaffronOlive's Budget Magic series for ideas and decklists. He posts decks for both Modern and Standard, so you can choose whichever format you'd like. And the list is huge-- there's something for every kind of play style. That said, keep in mind that Standard is a rotating format and is going to rotate again at the end of the month, so a lot of the Standard decks you'll find will have cards that aren't Standard-legal anymore. Modern is an eternal format though, so any deck you find is fair game (unless there's something listed that has been banned since then, but Modern bans are not that common.)
Modern's the format I play the most because it's the format with the most room for deckbuilding and thus the most budget options. You'll get a more "fair" game out of Standard due to it being a more strict format, but your card pool is super limited in comparison.
If you were interested in Modern, though, I'll suggest these decks for you to look at:
8-Whack - Super-hyper aggressive Goblin deck in which 90% of the non-budget version's cost is in 4 cards (4x Goblin Guide). The budget version is basically the exact same deck without them. Granted, you won't have the same early-game hits without them, but unless you're being really competitive I'd say you can find much cheaper options. This deck is explosively good and can put up a fight against the best of decks. I want to build it myself someday.
Black-White Tokens - This is kind of the equivalent of what Green-White Tokens is in Standard. Lots of tokens being dumped on the board, enchantments that boost them all. Of course, you can build the same concept with green cards instead of black, should you want some green pump spells instead.
Mono-blue Faeries - I make no secret of being a huge fan of Faeries in Magic. It's the perfect mix of aggressive play (note: all creatures in this deck are flying, so you'll often swing for unblockable damage) and control. I don't have the time right now to sit and explain all of the delightful mechanics, but watch the deck tech video for some more details. I love this deck and own it myself.
Tempered Steel Affinity - This is the Affinity deck I mentioned earlier that took me to 1st place before at FNM. I do like the deck a lot, but honestly I think 8-Whack is the better choice overall.
Turbo Fog Tutelage - This is a combo/mill deck. It's a pretty fun alternate win condition that's also a total troll deck for anyone who plays creatures to win. It doesn't do very well against decks that don't, however...
Zombie Hunt - The epitome of BUDGET. This is an combo-aggro deck that literally has 54 lands. Play Treasure Hunt to fill your hand, and play Zombie Infestation to discard all of those lands to replace them with an army of 2/2 zombies. It's a fun casual deck that is prone to MASSIVE blowouts. You will either win hard or lose hard. Mostly lose hard. But still, it's hilarious.
The next few are actual competitive decks that have expensive cards that can be replaced with less effective but still valid options.
Bogles - aka Aura Hexproof. This is essentially "Build a Monster: The Deck". Drop the expensive lands for cheaper ones and you've essentially cut the price of the deck in 1/3 of what it was. I wouldn't suggest cutting much else, though. But basically you play a small Hexproof creature and build it up with a bunch of strengthening enchantments. Since it's Hexproof, they can't kill it with targeted removal spells, and you're free to just stomp their faces in.
Soul Sisters - My personal pet deck and my main competitive deck. Drop the expensive lands (they are NOT vital). Still a little pricey for a budget deck, but it's delightful. Lots of lifegain and destructive power.
Stompy - The purest form of Green aggro. Cheap and competitive, you don't really need to drop anything.
Elves - More fun Green aggro, but revolves more around strength in numbers than pump spells. Drop the expensive lands and it's pretty damned cheap after that.
