It is important to note that Schoenberg didn't invent twelve-tone serial music specifically to get away from tonality or functional harmony. He had already been writing atonal music for more than a decade without using any kind of serialism. The problem he was trying to solve was that he struggled to give his atonal music unity, and he hoped to achieve that by keeping all 12 notes in constant circulation and by using a particular twelve-tone row as the underlying principle of the music. The twelve-tone technique has in fact more of a structural than a harmonic or melodic implication.
So the key to how to approach twelve-tone music lies in the techniques that Schoenberg, Webern and Berg had already been using in their earlier atonal music. They had developed a method of using small pitch-class sets as sources for motives. This didn't change; the twelve-tone row and its permutations provided a constant background of the same pitch classes, but the different versions of the row provided common pitch-class orderings, intervals and subsets, which could then be used as motivic source material.
As I mentioned in my answer to one of your previous questions, an important part of Schoenberg's, Webern's and Berg's twelve-tone music was their choice of twelve-tone rows. They didn't choose these randomly, but studied their properties, and used rows whose different permutations had shared elements that they could derive motivic material from. Below are a few examples:
Schoenberg: String Quartet No.4 op.37 (1936)
X:1
L:1/1
M:
K:C
%%score V1 V2
V:V1 clef=treble name="P0"
V:V2 clef=treble name="I5"
% 1
[V:V1] d ^c A _B F _E E c _A G ^F B
[V:V2] "_0"G "_1"_A "_2"c "_3"B "_4"E "_5"^F "_6"F "_7"A "_8"^c "_9"d "_10"_e "_11"_B
In this series there are four groups of three adjacent notes that form a pitch set with prime form [0,1,5]:
X:1
L:1/1
M:
K:C
%%score V1
V:V1 clef=treble
% 1
[V:V1] "_0"d "_1"^c "_2"A | "_2"A "_3"_B "_4"F | "_7"c "_8"_A "_9"G | "_9"G "_10"^F "_11"B |
This not only means that there are several instances of adjacent intervals 1 and 4 or 1 and 5, but also that many permutations will share groups of three notes; e.g. the I5 series-form shares the note group G-Ab-C with the prime form P0.
This is used by Schoenberg e.g. to link the first phrase (based on P0) played by the first violin, to the next phrase (based on I5) which is played by the second violin. Both times the note group G-Ab-C is played in the same register, and the note B which follows (almost) immediately is given a long duration; this makes it sound like the second phrase picks up where the first one left off.
Webern: Concerto for Nine Instruments op.24 (1934)
X:1
L:1/1
M:
K:C
%%score V1
V:V1 clef=treble name="P0"
% 1
[V:V1] "p0""_0"B "_1"_B "_2"d | "ri7""_3"_E "_4"G "_5"^F | "r6""_6"^G "_7"E "_8"F | "i1""_9"c "_10"^c "_11"A |
The series for this work, as often with Webern, is derived by taking a 3-note series, and then adding inversions or retrogrades of it to make up a full twelve-tone row. As a result of this, many series-forms contain the same groups of three notes (in order or reversed):
X:1
L:1/1
M:
K:C
%%score V1 V2 V3 V4
V:V1 clef=treble name="P0"
V:V2 clef=treble name="P6"
V:V3 clef=treble name="I1"
V:V4 clef=treble name="I7"
% 1
[V:V1] "p0"B _B d | "ri7"_E G ^F | "r6"^G E F | "i1"c ^c A |
[V:V2] "r6 (rev)"F E ^G | "i1 (rev)"A ^c c | "p0 (rev)"d _B B | "ri7 (rev)"^F G _E |
[V:V3] "i1"c ^c A | "r6"^G E F | "ri7"_E G ^F | "p0"B _B d |
[V:V4] "ri7 (rev)"^F G _E | "p0 (rev)"d _B B | "i1 (rev)"A ^c c | "r6 (rev)"F E ^G |
So even though the work uses many permutations of the complete twelve-tone row, the material is extremely limited, and the same intervals and note groups occur many times in different guises.
Berg: Violin Concerto (1935)
X:1
L:1/1
M:
K:C
%%score V1
V:V1 clef=treble name="P0"
% 1
[V:V1] "_0"G "_1"_B "_2"d "_3"^F "_4"A "_5"c "_6"E "_7"^G "_8"B "_9"^c "_10"_E "_11"F
This series contains four triads, their tonic a fifth apart like the open strings of a violin, and they appear as triads at several points in the work.
X:1
L:1/1
M:
K:C
%%score V1
V:V1 clef=treble
% 1
[V:V1] "Gm""_0"G "_1"_B "_2"d | "D""_2"d "_3"^F "_4"A | "Am""_4"A "_5"c "_6"E | "E""_6"E "_7"^G "_8"B |
Berg was not averse to tonal elements in his music, unlike Schoenberg and Webern, and especially the Violin Concerto often blurs the line between tonal and atonal, and even contains a Bach chorale as a literal quote.
As you can see, choosing the twelve-tone row for its combinatorial properties, and exploiting the possibilities of certain transpositions of the inverted and retrograde form, make it possible to create many connections between small motives in the work, and avoid what you describe as sounding like "random notes from the chromatic scale".
This also shows that the idea of all twelve tones being equally important shouldn't be taken at face value. The way Schoenberg, Webern and Berg use shared elements between different series-forms to generate motives creates temporary hierarchies between the notes, and a sense of foreground and background, of melodic motif and accompaniment, much like moving to the next chord in a tonal work changes the hierarchy between tonic and chord-tones and non-chord-tones.
Example
If you are asked to write a short phrase, or an unaccompanied melody, it is of course difficult to apply many of the techniques used by Schoenberg, Webern and Berg, and turn it into something interesting. However, the way Webern often derives twelve-tone series from shorter 3, 4 or 6-note groups, creates a microcosm in which you can use the repetition of intervals to link the parts of the series together. Consider this example:
X:1
L:1/1
M:
K:C
%%score V1
V:V1 clef=treble name="P0"
% 1
[V:V1] "^p0""_0"C "_1"E "_2"^F | "_3"_B "_4"d "_5"_A | "^i7""_6"g "_7"_e "_8"^c | "_9"A "_10"F "_11"B |
This series has four groups of three notes, and each of these pitch-class sets has prime form [0,2,6]. The first two groups combined and the last two groups combined form whole-tone scales, which are a transposed inversion of each other.
When turning the series into a melody, you can emphasize the shared elements between the different parts and make it sound far from random, e.g.:
X: 1
M: 6/8
Q: 1/8=110 "slowly"
L: 1/8
K: C
!p! C2 (E^F3) | _BD.!>!_A A2 (A | G2) (_E^C3) | =A=F.!>!=B B.!>!bb |
The use of the whole-tone scale provides the listener with a link to more familiar pre-twelve-tone music, and the (slightly dramatic) downward half-tone shift into another whole-tone scale brings the symmetry in pitch to the fore. The rhythm clearly sets the groups of three notes apart, and repeating the note Ab marks the end of the first whole-tone phrase (and helps to set apart the half-tone step to G), and then repeating the rhythm links the two phrases together. (I added the octave jump just for effect, and as a promise of what comes next.)