Is there a (preferably classical) Latin word for "time"? I mean "time" in the sense it has in "The first time is easiest." or "I can't remember the last time I was here.", not in the most common meaning of the word. In many other languages my kind of time is different from the usual kind of time, so I don't think tempus is a good translation (for all I know). The word I'm looking for is "volta" in Italian, "gång" in Swedish, and "kerta" in Finnish (not "tempo", "tid", or "aika"), if that helps. Latin has immensely useful numerals for this kind of thing (semel, bis, ter, …), but I am looking for a noun if one exists.
As far as I remember, Vicis is an option. It is far from a perfect match, though.
Spanish equivalent vez is derived from it.
L&S do not offer an nominative, but the Diccionario de la Lengua Española cites vix (as a noun, not to be confused with the adverb.)
L&S say (vicis, meaning I.B.1) it is Late Latin and cites as examples ager tertia vice arabitur (for the third time) and tribus per diem vicibus (three times a day.)
A couple of further examples:
Prima vice and secunda vice in the same context (court decisions in XVII century Naples.)
The usage has made it to the modern day legal expression pro hac vice for something that is permitted for this time/occasion only,
...and to botanics in a number of phrases/contexts.
Perhaps Joonas can say if this forum is of some help (translation of prima vice to Finnish.)
I would translate "The first time is easiest" as prima vice facilissimum est, note that you could arguably say prima vix facilissima est too, but with most evidence pointing to not having a nominative, the idiom seems to work better as a (time) complement, so I'm assuming there is something that is easiest when done for the first time. In the same sense, I'd translate "I can't remember the last time..." as something like Non memorare possum quando... novissima vice.
Finally I don't think this is a good term for just every occasion you want to say time with this meaning. Particularly for every time in some uses you can use quoties (and also as an interrogative in how many times.)
In medical prescriptions, up until almost the present day, doctors wrote b.i.d., t.i.d., etc. for bis in die, ter in die, etc., mean "two times a day", "three times a day", etc. This Latin grammar book by Leonhard Schmitz says that you can drop the preposition in, hence septies anno, "seven times a year". So, as you already noted, one way to render "a time" in the sense you have in mind is:
Use an adverbial number, like semel, bis, ter, quarter, quinquies, etc., and no noun at all.
That won't work when you want to modify a single "time" with some notion of ordinal number, though, as in "the first time", "the second time", "the next time", or "the last time". Surely we need a noun for that. Well, Smith's Copious and Critical English-Latin dictionary gives the adverb primum for "the first time". Also, saepius for "many times". The Vulgate provides some examples for other ordinal-ish concepts with "a time":
Et ait "Implete quattuor hydrias aqua et fundite super holocaustum et super ligna." Rursumque dixit "Etiam secundo hoc facite." Qui cum fecissent et secundo, ait "Etiam tertio id ipsum facite." Feceruntque et tertio.
And he said, "Fill four buckets with water and pour them over the burnt offering and over the wood." And again he said, "And do it a second time." When they had done it a second time, he said "And do the same thing a third time." And they did it a third time. (Kings 18:34)
This construction occurs many "times" in the Vulgate. So:
You can get ordinal "times" by using the ordinal adjective substantively.
But look at Judges 20:30:
Et tertia vice sicut semel et bis contra Beniamin exercitum produxerunt.
And they drew up armies against Benjamin a third time, just as they had the first time and the second time.
It looks like we've found your noun—some of it, anyway:
(missing), vicis. (f.)
Even here, Latin seems to prefer to avoid it, saying the equivalent of "once and twice before" for "the first and second times". Indeed, vice has no nominative singular! It seems to want to be in the ablative—where it anchors an adverbial phrase. However, since it's a noun, you can modify vice with anything you like. Here it is with alia, being used as a synonym for iterum:
Alia etiam vice Philisthim inruerunt et diffusi sunt in valle.
And another time the Philistines made a raid and spread themselves out in the valley. (Paralipomenon 14:13)
Deuteronomy 10:10 has "this time":
Ego autem steti in monte sicut prius quadraginta diebus ac noctibus exaudivitque me Deus etiam hac vice et te perdere noluit.
And I stood on the mountain as before, forty days and nights, and God heard me this time also, and he would not destroy you.
At ille subiunxit iuste vocatum est nomen eius Iacob supplantavit enim me en altera vice. Primogenita mea ante tulit et nunc secundo subripuit benedictionem meam. Rursumque ad patrem "Numquid non reservasti" ait "et mihi benedictionem?"
But he added, "Rightly is his name called Jacob, for he hath supplanted me lo this second time. He took away my birthright before, and now this second time he hath stolen away my blessing. Again he said to his father, "Hast thou not reserved to me also a blessing?"
I understand altera vice here as an emphatic substitute for iterum. Notice that in the next sentence, secundo denotes the same time, the fact that it's current emphasized with nunc.
So, while it sounds like vice is the word you're looking for, providing a noun to explicitly denote one time/occurrence, enabling you to attach modifiers to it ad libitum, the main thing I've gotten from digging for information about this is a striking reminder that in Latin, parts of speech other than the noun play a more important role than in English, often making a noun inappropriate or unnecessary.
With that in mind, here's my non-expert attempt to render your examples:
The first time is the easiest.
Facillime facere est prima vice.
The idea here is to avoid predicating anything directly about the first time (since there's no nominative singular), and instead make the action the subject and put prima vice in an emphatic position. Better still is probably Facillime facere est primum or even Primo est facillimum, "It's easiest the first time."
I can't remember the last time I was here.
Proxima vice hic adfuisse non reminiscor.
But that sounds very strange to me, more like "I can't remember having been here on the last occurrence." Probably better is Proximus hic adfuisse non reminiscor.
If you desperately need a noun in the nominative singular, you could reach for vicissitas or vicissitudo.
My French-Latin dictionary renders “fois” as “vicis”. Lewis/Short have for “vicis”, besides other things, this:
A time, turn (late Lat.; cf. Orell. ad Hor. C. 4, 14, 13): “ager tertiā vice arabitur,” Pall. 10, 1: “tribus per diem vicibus,” id. 1, 3 fin.; cf.: “tesserulas in medium vice suā quisque jaciebamus,” Gell. 18, 13, 1: “vice quādam,” once, Sid. Ep. 7, 1; Aus. Pan. Grat. Aug. 4.
There isn't a direct and simple alternative of the sense you are looking for to tempus, which is really a very flexible word and could be adopted into your examples. If, however, you find that tempus can't be used in precisely the sense that you require, there are other possibilities.
The meaning of locus is not restricted to physical location. It has its uses as a word for a point in time, e.g. ad id locorum, 'up to that point'; in illo locorum, 'at that juncture' (sc. of time'); this is probably the nearest answer to your actual question. You may prefer to use occasio, a convenient or suitable moment, etc., as in facillime ut primum occasio data est, roughly 'most easily at the first opportunity'.
It's not what you are asking for, but instead of trying to find an appropriate single word, you might find that temporal adverbs or conjunctions (cum, ubi etc.) can do the trick, as in hoc cum primum fecissem, 'the first time I did this'. Or straightforward circumlocution, such as salutationem nostram apud te nuperrimam obliviscor, 'I forget when I paid my last visit to you'.