Presumably you are well under age 70½.  If you had sufficient compensation in 2018 to support a $5,500 contribution for 2018, yes, you can make a $5,500 nondeductible traditional IRA contribution for 2018 by April 15, 2019.  It will be reportable on your 2018 Form 8606.  The result will be basis in nondeductible contributions appearing on line 14 that will transfer to line 2 of your 2019 Form 8606.

If you'll have sufficient compensation in 2019 to support a $5,500 contribution for 2019, yes, you can make a $5,500 nondeductible traditional IRA contribution for 2019 by April 15, 2020.  Note that the regular contribution limit for 2019 is $6,000, so you could actually contribute $6,000 for 2019 (given sufficient compensation).  The nondeductible contribution for 2019 will be reportable on your 2019 Form 8606.

Assuming that you have money in a traditional IRA, you can do a Roth conversion whenever you wish.  The Roth conversion is reportable on the tax return for the year in which the distribution from the traditional IRA occurs.  Converting immediately after you make the 2018 and 2019 traditional IRA contributions means that it will be reportable on your 2019 tax return that you prepare a year from now.

Had you made the contribution for 2018 before the end of 2018 and converted it to Roth before the end of 2018, the conversion would appear on your 2018 tax return instead of on your 2019 tax return.  Since converting immediately result in zero taxable income when you have no other money in traditional IRAs, there would be no difference in the taxable result had you contributed and converted $5,500 during 2018.

Once caveat to dmertz' great answer above.  You CAN owe taxes on traditional IRAs that are converted quickly before earnings accrue on new contributions  - IF you still own other traditional IRAs that have accrued earnings.  The IRS views your entire portfolio (collection) of traditional IRAs -existing and new - as one - and subject to taxation. Taxes would be proportional to the amount you are converting, relative to the value of your entire portfolio of traditional IRAs.

That is a good call out - I wasn't aware - can you elaborate?  I opened a new IRA to segregate post tax contributions.  I contributed post tax dollars, and then converted to Roth.  Independent of that, I had significant assets in a pre-tax IRA that had accumulated from rolling in prior employers 401Ks into that IRA.  Would the roth conversion trigger a tax bill based on the value of the pre-tax IRA?  I use turbotax and haven't seen any forms/documents to account for that.  

Under the tax code, there can be no segregation of post-tax contributions in IRAs.  There are no such things as separate "post-tax IRAs" and "pre-tax IRAs," only nondeductible contributions to your traditional IRA's in aggregate.  Your assets in all of your traditional IRAs are involved in determining the amount of any distribution or Roth conversion from any of your traditional IRAs.  For the purpose of determining the taxable amount of any distribution, your traditional IRAs are treated as if they are a single account.

 

This calculation is done on Form 8606 which TurboTax prepares automatically.  After entering a Form 1099-R reporting a Roth conversion, you must click the Continue button on the Your 1099-R Entries page and enter your total balance in traditional IRAs at the end of the tax year so that TurboTax can accurately prepare Form 8606.