If I am receiving a $35,000 enlistment bonus, what is the actual amount of the bonus after taxes are deducted?

The 25% figure CPT Lawrence Cable gave is always a great estimate to use. It will give you a high-end figure almost always, but it's better to be pleasantly suppressed than the other way.

If you want to sit down and do the math for a more accurate figure, you'll have to figure out the following:
● First, is this for 2023 or 2024? If for 2023, then it's pretty easy - you don't have to guess, just plug in what you know - go to mypay.dfas.mil and use the W-2 info (if available) or the info from your December LES.
● What's your total income going to be for the year? Are you going to be a PFC the entire year or do you expect to be promoted to SPC at any point? For the upcoming year, just estimate how many months you would be each grade, compare to the respective pay tables*, and multiply the # of month by the monthly salary.
● As part of that income, are you getting the entire enlistment bonus in a single year (you are taxed based on the year you received the income)? For a $35,000 bonus, typically it would be half up front and the rest over the term of your enlistment.
● Do you claim a state that has income tax as your HOR (there are nine states that don't tax your income), then you'll have to add even more onto that. For example, if it is California that you'll have another 13.3% for state taxes .. if North Dakota, then it would be an addition 2.9%.
● How much are you contributing to your TSP under the blended retirement system? By the way, if you're not putting AT LEAST 5%, then you are short-changing yourself as the government will max it's matching contributions at that point.

Take all that and plug it into one of the online tax estimators*.

For an example, I'll assume this is for 2023, you were a PFC the entire time, you are being paid under 2 years, You just enlisted last year and started on 1 JAN 23, you received 1/2 of your bonus up front, you put 5% of your pay in your TSP, you have no additional income, and your HOR is Oregon.

Plug all that into the tax estimator and it comes out with an effective tax rate of 21.32% .. pretty dang close to the 22.55% figure CPT Lawrence Cable had before he rounded up to 25%.
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* 2024 Military Pay - https://militarypay.defense.gov/Portals/3/Documents/ActiveDutyTables/2024%20Pay%20Table-Capped-FINAL.pdf
* Tax Estimator - https://smartasset.com/taxes/federal-tax-calculator

I am commenting only to state that questions like this one, along with the detailed responses, showcase the valuable purpose of this forum for Servicemembers.

I used the 22% tax amount when I counseled Soldiers, all depends on your tax situation. Now, keep in mind, the $35,000 will be broken down based on the number of years on your enlistment contract, you will not get the entire $35,000 all at once.
Right after you graduate AIT and report to your first duty station, you will submit the paperwork to finance and get 50% of that amount, $17,000 pre tax, the remainder of the bonus will be paid over the anniversary year of your contract. Lets say you have a four year contract:
Grad AIT/report to first duty station, $17,000,, leaves $17,000 left to pay out.
Year 1=$4250
Year 2=$4250
Year 3=$4250
Year 4=$4250

All those numbers are pre-tax amounts. Even if you deploy to a tax free zone, your payments will still be taxed based on the fact that the original contract was done in a taxed status.
Keep this in mind:
When you get to your first duty station, you need to have your entire enlistment contract, Annex A and Annex B of the contract, your graduation certificate (AIT) and the orders awarding you the MOS in order to the bonus to be paid.