Why did the people want a king? Why couldn't they just find another shofet? Can't also shofets lead in battle (Joshua, et al.)?
The people wanted a king so that they could more closely resemble the other nations (Sh'muel I, 8:5,20; Radak ad loc.). This motivation made their desire for a king contemptible (Sanhedrin 20b; Sh'muel I, 8:7-8; cf. D'varim 17:14-15), despite the fact that, according to some opinions, there is a biblical obligation to appoint a king (see the dispute in Sanhedrin 20b; Rambam Hil. M'lachim 1:1 follows the view that it is an obligation).
The Ramban comments (D'varim 17:14) that although the verses in D'varim indicate a command to appoint a king, the verse merely predicts that the people would want to be like the other nations. Such a desire is not part of the command to appoint a king, nor is it praiseworthy.
The Talmud (ibid.) notes that the passage in Sh'muel I indicates competing motives in asking for a king; the elders properly requested a powerful authority figure to replace Sh'muel and judge the people after Sh'muel's own sons proved unworthy (8:5), while the masses (whom the Talmud calls "ignoramuses") erred in wanting a king to judge them and lead them in battle so that they would be more like the other nations (8:19-20).*
The general question of why a king might have been considered more suitable than a shofeit (loosely translated as "judge") for waging war is a good one. See the Abarbanel's introduction to Shoftim, where he enumerates five similarities (perhaps not all undisputed) between a king and a shofeit, including their leadership roles in battle and their command authority over all Jews. This question deserves an extensive treatment. Briefly, I think one approach is that judges did not always have the practical ability to unite all the tribes behind them in war. Two examples of this are Devorah (Shoftim 5:14-18) and Gid'on (ibid. 7:23). An annointed central figure would have undisputed authority among all Israel, and would thus be better suited to organizing the entire nation's efforts in battle.
*According to this approach, the phrasing of the elders "שִׂימָה לָּנוּ מֶלֶךְ לְשָׁפְטֵנוּ כְּכָל הַגּוֹיִם" may be taken as explanatory, while the phrasing of the masses "וְהָיִינוּ גַם אֲנַחְנוּ כְּכָל הַגּוֹיִם... וְנִלְחַם אֶת מִלְחֲמֹתֵנוּ" suggests a desire to imitate the other nations as well as a lack of faith in the Almighty's ability to lead them to military victory. Alternatively, as perhaps suggested by the negative response to the original request, the first request "שִׂימָה לָּנוּ מֶלֶךְ לְשָׁפְטֵנוּ כְּכָל הַגּוֹיִם" may have come from the people as well as from the elders, and the elders might not have been the ones who stated any comparison with other nations.